One might, on the other hand, wonder what might have happened if the Mongols hadn't been infected by Islamic schism and dynastic troubles and completed their push through into Mamluk Egypt. As pivotal as Ayn Jalut was, the groundwork for that failure was laid in the religious conversion of [1]Berke and the Golden Horde to Islam.
[1] Initially listed his brother Batu, rather than Berke, the first Muslim Khan.