ilkhanate

Simple enough question: how possible would it have been for Mongol invaders (either in the form of the Golden Horde or the Ilkhanate) to have seized Constantinople during the 13th century? What might be a plausible series of events leading to this? Depending on when such a siege would occur...

I don't want to start a discussion about if the Manchus were chinese or not, but most of the modern chinese dislike a lot the Qing dynasty since they were ruled by manchus and not by Han, period. Now, now, on a scenario were the Ilkhanate survived further than the 1600s AD, could some kind of...