What if the Mongols had continued their attack in the Middle East? Could they have conquered it, at least temporarily?
Even if the Mongols fail to take and hold Egypt or the Levant, a victory at Ain Jalut would probably mean a more secure, longer-lasting Ilkhanate in Persia and Mesopotamia.
Why so? Is the Ilkhanate going to be better lead or more accepted or otherwise internally better off?
I almost wonder if this is one of the "turning points that failed to turn."
Hulegu, as Elfwine pointed out, was not present. Moreover, the Golden Horde ceased sending reinforcements to Hulegu, and the two actually began fighting.
I'm not sure I'd count out Egypt, either. The 5th Crusade shows the danger of assuming Egypt is supine and easy to conquer. Why can the Mongols take Damietta when the Crusaders couldn't?