Its very simple to deal with the Mongols. First Understand that its impossible to kill them on the field, because:
1. All their Units are Pumped up to high skilled warriors, meaning their peasants will beat the crap out of your peasants.
2. Their Horse Archers are the elite and the backbone of their army. They will outrun you, Outfight you, outshoot you and outkill you.
Roughly One of their units is equivalent to two of yours.
So never engaged them on the field. Unless you happen to have your army made of thousands of Spearman and Archers fighting from the top of the peak that has only one way up. This is (while happening to me once) barely going to happen.
So always engaged them in a city. Fill up the barracks with maybe 8 Missle Units, 5 Spear/Really REALLY STRONG units, 3 Heavy Infantry and the rest Artillery or Calvary of your choice.
Also remenber to upgrade each City with the most advance fortifications, meaning Cannon on walls and towers.
Even if you are facing 2 Full Armies to 1 full of yours, and you know you are sure to lose, dont give up and auto-fight. Fight to the last fricken man. You usually can kill more than an Auto-fight can do. Sometimes you might just win dued to luck (Death of Khan early in battle), Glitches (Two armies assualted the city, one of them did not move) and also some skill (Jamming Spearman into the gate as their Horses try to squeeze into the city with missle archers firing those still outside the city.)
Dont be afraid to lose a city, just use the garrision inside kill the invaders, then while their weak, find the chance and strike back. The win might be short live as they will strike again but every Mongol you kill, is one you wont have to kill later.
Also dont crowd A lot of full armies around one city as the invading army would just go around your heavily fortified city to a weak one.
Thats all the advise I can give, with any luck, you will end up with these results:
Same City Sieged three times
First Siege, won with the combination of Skill, Luck and Glitches (read above for explaination)
Second Siege, Won again much like the first sieged, killed off their King too
Third Siege, Finally Lost but killed about 250 more people than I had. Also killed their king and one of the relatives
Good Luck
Also ask Steph. I think he knows how to deal with them too
There are also strategies like holding them off at the rivers in Iraq and Killing off the smaller armies before the regroup to be a big 4 menace