Seeing how you have a large empire and thus likely many castles, I would just stall them for a while by building a fort or two they can waste time on while you raise several strong armies nearby.

As for composition, you want to have a lot of missiles (pavise) and spearmen (sergeants) to defeat them. Heavy fast cavalry is useful too, so you'd want to get several mailed knights for taking out their archers and imperial knights for holding their cavalry.
Don't really need artillery against Mongols.

Your advantage is terrain really, as their units get penalties for fighting in woods and grassland while yours get a bonus. So it doesn't matter where you engage them.
Some will say that you should hold bridges and defend against siege, but tbh, that is not necessary and costs you a lot of upkeep in the long run if they stay relatively idle, which they tend to do.

Just pick a general or two and send them running back to the east.