Actually, several academic books on the Mamluk Egypt have recently published, not only from the dedicated academic series like Mamluk Studies (Vandenhoeck & Ruprecht), in the last decade.
A specialized academic journal, Mamlūk Studies Review (U of Chicago) is also open access, so you can check the book review section of individual issues of this journal to find more dedicated new literature.
On the other hand, we might certainly have some difficulty in the good introductory on them. I suppose the following books look fine (and I have used some of them for research before), but there might be more suitable books to be noted here:
- Islaeli Scholar, Reuven Amitai or Reuven Amitai-Preiss publishes several books focusing on the relationship between the Mamluks and the Mongols, like Holy War and Rapprochement
Studies in the Relations between the Mamluk Sultanate and the Mongol Ilkhanate (1260-1335). Turnhout: Brepols, 2013.
- Thorau, Peter. The Lion of Egypt : Sultan Baybars I and the Near East in the Thirteenth century, trans. P. M. Holt. London: Longman. 1992: is a revised translation of the reputed German biography of Baybars.
- (Added) I forgot to mention the classic overview work, P. M. Holt, The Age of the Crusades
The Near East from the Eleventh Century to 1517, London: Routledge, 1986.
Difficult to find books on 'Mamluk Egypt'? Then, I'd recommend to change your search word to 'Medieval Cairo'. There are many urban history books on medieval Cairo, and most of them actually focus not on the Fatimid and Ayyubid dynasties, but on the Mamluk period.