According to Wikipedia the Mamluk Sultanate ruled part or all of the territory of the following countries:

> Today part of

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 Egypt
 Israel
 Jordan
 Lebanon
 Libya
State of Palestine Palestine
 Saudi Arabia
 Sudan
 Syria
 Turkey

Thus the maximum area of the Mamluk sultanate must be smaller than the total area of those countries.

Egypt: 1,010,407.87 square kilos

Syria: 185,180 square kilos

Israel: 20,770 square kilos

Jordon: 89,341 square kilos

Lebanon: 10,452 square kilos

Palestine: 6,220 square kilos

total area: 1,322,370.87.

The Mamluk Sultanate probably ruled most of the area of those countries.

Then there are four large countries that the Mamluk Sultanate probably ruled relatively small parts of:

Libya: 1,759,541 Square kilometers

Turkey: 783,356 Square kilometers

Sudan: 1,861,484 Square kilometers

Saudi Arabia: 2,149,690 Square kilometers

total: 6,554,071 square kilometers. But the Mamluk Sultanate ruled relatively small parts of those countries.

Thus the Mamluk Sultanate probably ruled somewhere between about 1,322,370.87 square Kilometers and about 7,886,441.87 square kilometers.

The Mamluk Sultanate in 1337 was approx. 1 802 679 sq km.

The Mamluk Sultanate in 1444 was approx. 838 316 sq km in directly ruled territories plus 354 751 sq km governed indirectly without Cyprus, an additional 9 251 sq km, which from 1426 to 1473 was at least under nominal Mamluk suzerainty if not an official dependency.

See maps towards the end of this post with regards to what this territory included for these dates.

Accuracy & Caveats

This answer was reached by referencing Paradox Studios' 'Crusader Kings II' (CKII) and 'Europa Universalis IV' (EUIV) starting maps for 1337 and 1444, georeferencing these, and measuring the georeferencing area.

Some problems should be expected using this methodology:

• It relies on the accuracy of Paradox's map-making;
• It relies on the accuracy of Paradox's historical modelling and research;
• Inhabitable territory in deserts may have been extended outwards compared to the real situation to improve gameplay;
• Minor territories may have been excluded from these maps;
• Small enclaves / exclaves are probably not represented due to the scale of the map;
• Georeferencing can introduce errors.

I was lucky to find an already georeferenced map of all of EUIV's provinces, and I was able to assess some of the accuracy there:

• Cyprus, which is 9,251 sq km, is a province of 9,202 sq km (99.4% accuracy);
• My "drawn" Mamluk Sultanate (1444) amounted to 963 583 sq km; the measured provinces on the pre-georeferenced map to 903 663 sq km (an 'overestimation' of 6.3%). => Edit: Both of these numbers were quoted prior to me removing the Eastern Desert as prompted in the comments.

In light of the above, I would give the 1444 area value an accuracy measure of ±2-5%.

Regrettably, the CKII map was a lot more difficult to georeference accurately (because of its bounds). Only in Cyrenaica, I might be overestimating its area by about 10 086 sq km. The rest of the territory (towards Asia) worked a lot better though, so I would give an accuracy measure of ±4-8% for this date marker.

Maps

So everyone can see the territory the games cover at these dates, I've included snippets below. I have removed the same "Egyptian Desert" (reflecting the Eastern Desert) area from both sets of maps.

1337 Game Scenario:

The territory under the "Bahri" is depicted here:

1444 Game Scenario:

Territory under the Mamluks is what I've considered as "directly ruled" above while the "indirectly" covers the Fadl, Hejaz, and Medina which are depicted as (at least relatively) independent nations in this timeframe.

1337 Comparison:

To see the "accuracy" of these maps against the GSHHG coastline (and rivers and lakes) and in comparison to each other, I've added two further comparative maps below.

1444 Comparison: