As someone said before, the factions with strong infantry and mounted skirmishers and lancers are the best for the Mongols. That would mean the Turks, Egyptians, Byzantines, and maybe the Russians in that order. Oddly enough, those are also the 4 factions on the front lines of the Mongol invasion. It's just a matter of how many forces you have to spare to fight them with without your front lines collapsing elsewhere.
The Turks and Egyptians also have a very unique advantage against the Mongols and Timurids that nobody else gets. When it almost comes time for the Mongols to arrive, Elephants and Elephant Artillery will be avaliable as mercenaries in the Baghdad/Yerevan region. So you can have some of their secret weapons to use against them when they arrive.
Any faction can recruit them as they are mercs, but the Turks and Egyptians start closest to that region.
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