The perspective of Russian makes me wonder if the Mamelukes could only muster this one mid-size army (approx. 15k) and only had one throw of the dice, or whether (or not) they had resources elsewhere?
Let's see at power base of Egypt.
It does not even matter which dynasty would be in charge in Egypt after Ain Jalut. The Mameluks were the core of Egypt army for a long time before they themselves took power.
- the local Egyptian population (ethnic Egyptians) had not served in the army for about a thousand years or so, it would be a bad idea to conscript them right now, especially against the best army in the world, the Mongols. There is such thing as military tradition, which cannot be built in a few months or even years. You cannot take a local peasant, give him a spear, explain how to stick it into an enemy and hope that he will be a match for a Mongol. Actually poor martial qualities of the local population forced the authorities to buy slave warriors.
- 95% of all the Egypt Mameluks took part in the battle. There was about 5% left in Egypt which were mostly sick and second rate warriors. We know that essential feature of the Mongol warfare was to eliminate as many enemies after the battle as possible. So we may make an educated guess that there would be quite few Mameluk survivors after the unfortunate battle.
So how could you replenish Egypt's army with Mameluks? You cannot just go to the nearest local market-place and buy yourself ten thousand slave-warriors. First of all it was not any male slave who could be made a Mameluk. The traditional areas of Egypt's Mameluk slave "recruitment" were Turkic Cuman tribes, Georgians, Armenians, Circassians and some other Caucasian tribes. In OTL it was a very serious problem, even an existential threat to Egypt when Il-Khans tried to cut Egypt away from these traditional recruitment areas.
But even if we imagine that Egypt could buy slaves suitable to be made Mameluks they need at least 3-5 years to form an army, a fighting force to be reckoned with. And it was not only about military training which in itself was essential, it was the intricate system of specific loyalties of the warrior-slaves to their new masters, to Sultan, to their Mameluk caste as a whole, to Egypt protecting which was their main responsibility - these were the features which made them real Egyptian Mameluks, not just an unruly bunch of slave cut-throats.
Could Egypt buy a ten thousand new slaves suitable to be made real Mameluks? - Nope.
Did Egypt have these 3-5 years to train them? - Sorry, but no.
- There were warlike Arab tribes living on the borders of Egypt and some Turkoman tribal refugees who fled from the Mongols. They joined the Egypt's Mameluks against the Mongols as they hated them and hoped that after withdrawal of the main Mongol forces they had a good chance. But now after the defeat those Arabs and Turkomanns who survived - seriously regretted that they had not smelled the winner.
Even if some of them had joined the Egyptian resistance they would have been the serious destabilizing factor as they were alien to the political system of Egypt, their loyalties were along different lines than those of the Mameluks.
So after Mameluk disaster at Ain Jalut Egypt would be without any army able to resist the Mongols in the open field.
There would be political turmoil in Egypt, infighting.
The cities desperately trying to defend on themselves, open countryside - that is the favorite situation for the Mongols, that is where they belong. That is their favorite pastime, like hunting - moving from city to city looting countryside and giving the choice to the cities - to be burned to the ground or open the gates to their new true masters.
By the way, that where the Crusaders might get handy - like Christian Armenians and Georgians they might be trusted and used as auxiliary military force against the local Muslim population.