I'm just finishing off the Mongols in my Egyptian campaign and went through a similar battle with them. I've waged many battles against them and noticed that there were some similarities when I won.

The key for me (I'm playing VH/VH, so it might be amplified a bit) was having the high ground. I would save before the attack, and try each of the attack angles (I justify this by thinking of it as scouting. Any real general would scout the lay of the land before an attack, not my fault CA didn't impliment that).

I made sure to have at least two full stacks attack at once. Unfortunately, you have to put 1 on AI control. I set my troops on any hill I can find, so that there is a long uphill between me and the Mongols. They send their HA first and I stay on the hill, firing down with merc crossbowmen (imported from Italy), desert archers, and mamluk archers. As they move their infantry forward, my AI-controlled stack is slowly moving forward. I split my HA into two and send them out to the wings and slowly flank to the back. I send desert calv (aka assassins) to the wings to get into position to take out bodyguards/heavy lancers. When their archers get close enough to fire on my stack, I send calvary (a arab calvary/mamluk/bodyguard mix) to hit them. After the initial charge, I pull the calvary back and let my spearmen/infantry (sarcaren militia mostly, some ghazis in there too) into the fray. The key now is to circle the calvary around to charge into the back of engaged troops. Mongols have HUGE moral, so don't expect a route as quick as another faction, but it will happen.

The other AI stack should arrive at about the same time as your initial calvary charge. While they won't be nearly as useful as if you were controlling them, they'll certainly help wipe up the fleeing Mongols. Be very aware of Mongol reinforcements. You might have shattered the first wave, but the 2nd mongol stack will show up at the same time as your reinforcements and can hit you when you're spread out.

Sorry for the long post. There are some good posts in the history if your search in the Citidel too. Good luck!