How does the LDS Church reconcile their understanding of the "Godhead" which is three beings/three gods with the Bible stating there is only one God?

The LDS Church teaches an important belief...

We believe the Bible to be the word of God as far as it is translated correctly; we also believe the Book of Mormon to be the word of God. (Article of Faith #8)

As you ask your question, please remember that we simply do not agree that the Bible teaches the Trinity is in any way a single entity. You've presented a number of verses that would suggests that the Trinity is but one entity, but only if you begin with a belief that your perspective is true.

And now I am no more in the world, but these are in the world, and I come to thee. Holy Father, keep through thine own name those whom thou hast given me, that they may be one, as we are. ... That they all may be one; as thou, Father, art in me, and I in thee, that they also may be one in us: that the world may believe that thou hast sent me. (John 17:11,21 (emphasis mine))

While I believe a discussion of Jesus' baptism is more dramatic (explaining why God would want to deceive His children by appearing to be three entities — the voice of the Father from the heavens, the body of the Son in the water, and the Holy Ghost in the sign of the dove — is always interesting), the above verse most clearly supports our belief from a Biblical perspective. (Curiously, you allude to this when you reference John 10:30.)

• We believe Jesus is NOT teaching that He, the Father, and His disciples are a single entity separated physically in mortality but coexisting as a single entity eternally. And yet He states that He wishes them to be "one" just as He and the Father are "one." Unless you want to believe in (at least) a quindenity (15-in-1), the scripture can only be rationally interpreted as "one in purpose."

And that means that, unless you have a verse that clearly teaches, "there is no Father, no Son, nor a Holy Ghost, there is only God" (IMO a very Judaic perspective), all other verses suggesting that the three are a single entity must be viewed through the glasses of John 17:11 and 21.

Colossians 2:9

Let's review the verses you mentioned. I had to do a bit of checking on the first one, because Col. 1:19 doesn't say what you said. I suspect you're referring to Col. 2:9.

For in him dwelleth all the fulness of the Godhead bodily. (Colossians 2:9)

We believe there's a difference between "the fullness of the Godhead" and "the Godhead." This verse does not say that the Godhead is bodily within Christ, only that the fulness of the Godhead is.

And of his fulness have all we received, and grace for grace. (John 1:16)

That Christ may dwell in your hearts by faith; that ye, being rooted and grounded in love, may be able to comprehend with all saints what is the breadth, and length, and depth, and height; And to know the love of Christ, which passeth knowledge, that ye might be filled with all the fulness of God. (Ephesians 3:17-19)

That "fullness" is something that each of us can also have dwell within us — and yet I suspect few believe that all we mortals on Earth are a single entity, existing both here and in Heaven simultaneously. It could be argued that the use of the word "bodily" makes Col. 2:9 something different. But does it?

The Greek σωματικῶς appears to only identify where the "fullness" dwells. But if that is unique to Christ (as expressed in the tenet of the Trinity), then in the two verses I quoted, where does the "fullness" dwell? John said he and the disciples received it. Paul taught that it dwells in our hearts.

But ye are not in the flesh, but in the Spirit, if so be that the Spirit of God dwell in you. Now if any man have not the Spirit of Christ, he is none of his. And if Christ be in you, the body is dead because of sin; but the Spirit is life because of righteousness. But if the Spirit of him that raised up Jesus from the dead dwell in you, he that raised up Christ from the dead shall also quicken your mortal bodies by his Spirit that dwelleth in you. (Romans 8:9-11)

If the Spirit of God dwells within me, and the Spirit of God is intrinsically part of the Godhead, am I not therefore part of the Trinity?

Obviously, no. Colossians 2:9 only teaches that Jesus embodied the full purpose of the Godhead.

Deuteronomy 6:4

Hear, O Israel: The Lord our God is one Lord (Deuteronomy 6:4)

Remember when I mentioned that some of the verses you quote only make sense the way you say if you look at them with a predisposition to believe they're talking about the Trinity?

Latter-day Saints believe that Jesus Christ is Jehovah because it is true. It was known to be true anciently, by prophets in both the Old and the New Worlds, and it has been made clear from the very beginning of our own dispensation. The Lord Jesus Christ himself has declared it unequivocally: he is Jehovah, the God of the Old Testament. (Source)

The LDS Church believes that Jesus and Jehovah were and are the same person. Different names, same person. You do, too, if you think about it, because if the doctrine of the Trinity is true, then Jesus and Jehovah must be the same people intrinsically.

But what's really curious is that even Judaism doesn't universally believe that verse implies monotheism:

The precise meaning of the Shema is uncertain. The four Hebrew words “YHVH eloheinu YHVH ehad” literally mean “YHVH our God YHVH one.” Since Hebrew does not have a present‑tense verb meaning “is” to link subject and predicate, the link must be supplied by the listener or reader. Where to do so depends on context and is sometimes uncertain. Grammatically, “YHVH our God YHVH one” could be rendered in several ways, such as (1) “YHVH is our God, YHVH alone”; (2) “the LORD our God, the LORD is one” (lit. “YHVH our God, YHVH is one”); (3) “YHVH our God is one YHVH.”

An interesting perspective is given in the article's prologue, "Dr. Stephen Geller of the Jewish Theological Seminary understands the word 'one' to imply superiority of power–as in, 'YHVH is #1'!–rather than as a statement regarding monotheism." Please remember my first citation, "We believe the Bible to be the word of God as far as it is translated correctly."

We believe that the verse reasonably expresses a valuable truth: that Jesus Christ is the God of this world. Not the devil. Not Zeus nor any other god. He stands separate and under the authority of the Father, but this world is His.

For there is one God and one Shepherd over all the earth. (1 Nephi 13:41)

There is none other way nor name given under heaven whereby man can be saved in the kingdom of God. (2 Nephi 31:21, read that complete verse to discover that many prophets expressed the idea of unity of purpose in the same way.)

It is LDS doctrine that Jesus Christ was given authority from the Father to create and administrate the Plan of Salvation for the Earth. Therefore, the statement in Deuteronomy is entirely correct — when translated correctly.

Isaiah 44:6,8

I believe you've taken this verse out of context. let's grab a bigger chunk so we can see what that context is:

(6-8) Thus saith the Lord the King of Israel, and his redeemer the Lord of hosts; I am the first, and I am the last; and beside me there is no God. And who, as I, shall call, and shall declare it, and set it in order for me, since I appointed the ancient people? and the things that are coming, and shall come, let them shew unto them. Fear ye not, neither be afraid: have not I told thee from that time, and have declared it? ye are even my witnesses. Is there a God beside me? yea, there is no God; I know not any.

(9-11) They that make a graven image are all of them vanity; and their delectable things shall not profit; and they are their own witnesses; they see not, nor know; that they may be ashamed. Who hath formed a god, or molten a graven image that is profitable for nothing? Behold, all his fellows shall be ashamed: and the workmen, they are of men: let them all be gathered together, let them stand up; yet they shall fear, and they shall be ashamed together.

It would be nice for everyone on the planet if Deity would simply write a concise explanation in the form of a textbook concerning all aspects of Heaven and Earth. They didn't. They followed some pretty cool rules that guarantee everyone has Free Choice — which includes working primarily through intermediaries like prophets.

What was the Lord saying through Isaiah? That compared to the invented gods of the world, there is only one god, Jehovah.

Which we've already proven is Jesus.

Who members of the LDS Church believe is one in purpose and effort with the Father, the God of this world, and the only God through whom we will receive salvation.

So the statement made in Isaiah 44 is not a statement of monotheism — in fact it's another statement like Deutoronomy 6:4. Jehovah is #1! We agree with Isaiah 44:6-8 entirely. We believe there is no god of this Earth, factually in the form of the devil (who pretends to be an angel of light) or fictionally in the form of Zeus, Ishtar, Lir, or any other. We do not believe this statement excludes the Father because Jesus was made the God of this Earth, and it is through Him alone any are saved.

Mark 12:29-31

I believe you're quoting Mark 12:29, not Mark 10:29.

And Jesus answered him, The first of all the commandments is, Hear, O Israel; The Lord our God is one Lord:

And thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy mind, and with all thy strength: this is the first commandment. And the second is like, namely this, Thou shalt love thy neighbour as thyself. There is none other commandment greater than these. (Mark 12:29-31)

First of all, Jesus is quoting Deuteronomy. His original language would have been Hebrew, not ancient Greek (much less 17th century English). A fair argument could be made that Jesus, who knows better, would provide a more accurate translation. Except that it wasn't Jesus who wrote or translated the book of Mark (there's that pesky "as far as it is translated correctly" problem again).

Additionally, I quoted verses 30-31 because, once again, we need the context of the verses. Jesus is trying to make a point about people's behavior, and he's using a statement they would be very familiar with from the Torah to make it. There is only one God who can save you (1 Nephi 13:41; 2 Nephi 31:21), and you should devote yourself to Him, and to each other just as you would Him.

Once again, Jesus' statement was not a statement of monotheism. It was a statement inviting people to focus on a truth the LDS Church completely agrees with.

TL;DR

That was lengthy, but it needed to be. What you were really asking was, "how does the LDS Church reconcile it's beliefs with mine?" Honestly, we don't need to anymore than you need to reconcile yours with ours. The purpose of this Stack is to discover understanding, to learn about what other people believe and why they believe it. To that end, it's important to understand what each of us brings to the proverbial table.

LDS Historian Ben Spackman once said,

Each of us has in our head, a black box full of presuppositions, cultural assumptions, and worldview. Into this black box goes the text of scripture, where it interacts with these unconscious things in our head, and out comes “what scripture says.” The contents of that black box vary from person to person, so people with different presuppositions and worldviews will read the exact same text, and come away with very different understandings of what the text means. (Source)

Members of the LDS Church bring their views of the world, their beliefs in more of the Word of God than is found in just the Bible, and their theology, to interpretation. I can list dozens of scriptures from our Canon:

• The Bible
• The Book of Mormon
• The Book of Doctrine and Covenants
• The Pearl of Great Price

I can also list dozens (if not hundreds) of citations from Church leaders and authorities, all expressing and explaining our beliefs. The question on this forum is not supposed to be "are we right?" The questions are only ever supposed to be, "what do believe and why?"

Thanks for asking your question!

In D&C 20: 28 we have

Which Father, Son, and Holy Ghost are one God, infinite and eternal, without end. Amen

Mosiah 15:1-5 has a discussion of how the Father and the Son are one God. The resurrected Christ declares in 3 Ne 11:14

...I am the God of Israel, and the God of the whole earth

but later in 3 Ne 11:36

The Father, and I, and the Holy Ghost are one.

Yet He prays to and speaks of the Father as a distinct, separate being than Himself, as He did throughout the New Testament.

The LDS reconcile these apparent contradictions by understanding the unity of the Father, Son, and Holy Ghost differently than most Christian theologians and philosophers would have it. It is not understood to be an ineffably mysterious identity of being or substance. It is rather understood to be a perfect unity of mind, outlook, understanding, purpose, will, and power. There are no quarrels, no wrangling, no cross-purposes, no separate agendas (as, for instance, the Greek or Roman gods).¹

The LDS also take note of John 17:20-23 where Jesus prayed that those who believed would also become "one, as thou, Father art in me and I in thee", which points to a different meaning of "one" than is usually understood in Christian theology.

_{1 Gospel topics: godhead}