I was thinking about this today. I'm sure this has been discussed before on the board, and if anyone knows of any threads to link to, I would appreciate it.
The truth is that off the top of my head, I don't know if they could have done it.
When I say, "Defeat the Mongols" I either mean that they make it so that the Mongols remain in their own sphere of influence, without conquering either empire (or without conquering the Song), OR the Mongols are crushed, which obviously would have been much harder and unlikely.
Were there any tactics in particular to deal with the Mongols unprecedented mobility, and their amazing horse archers? Aside from other horseman, traditionally spearmen are the best for dealing with horses. But the Mongols were always able to simply work at a distance from their enemies.
There were also the very effective advantages the Mongols had in their brilliant leaders, good army organization, and other advantages.
Since the Mongols seemed to have difficulty whenever there were issues of succession, maybe assassins could have screwed with the chain of command enough so that the Horde became too divided once again? I'm sure many missions like that were attempted, though.
What are your thoughts?