If I had to meet them on the field (Ugh! Note: I have never done this against Mongols, but am basing this off of scores of like battles against other factions in M2TW and in RTW) I would do at least one of the following three things:
1.) Build 2 or more full stacks of Vardariotoi and/or Kataphractoi, which will cost you about 32000 plus 10000/turn in upkeep. (Here is where your good economy comes into play.) Vardariotoi are roughly equivalent to the Mongol Heavy Archer; and Kataphractoi, if they can force a melee engagement, look slightly better at hand-to-hand combat. Approach Mongol stacks from uphill. Park one stack next to a Mongol stack and attack with the other. If you can't get at isolated Mongol stacks, try to kill off the first stack before enemy reinforcements arrive.
2.) Need a cheaper option? Build 2 or more full stacks of: 10 Spear Militia + 2 Varangian Guard + 8 (Trebizond or Byz Guard) Archers. This time, gaining high ground is more important. Try to get attacked on the campaign map (rather than attacking). Form a defensive spear line in a corner of the map or against a mountain or other natural obstacle. Put spears on guard mode. Have your V.G. in loose formation behind the spear line and ready to reinforce. Have your archers all in loose formation in the protected pocket, with scirmish mode = "off" and set to "fire at will". Sit back and have a snack as you watch the battle. Stop snacking and send your V.G. into action if your spear line starts to fold.
3.) An even cheaper option, if the geography allows it: Build one stack of: 10 Spear Militia + 2 Varangian Guard + 8 (Trebizond or Byz Guard) Archers (just like one of the stacks from option 2). Use it to control a bridge.
Again, getting attacked on the campaign map is key to victory on the tactical map.
Your archers have longer range than the mounted archers. Place them off to your left flank, where they can shoot into the enemy's right (unshielded) flank. Leave them on loose formation. Put your spears in a 'U' formation at the end of the bridge. Form the 'U' such that the first enemy unit can come almost completely off the bridge and get spears poked into them from the front and from both flanks. Pack other units in behind your 'U'. If the enemy stack includes infantry, you might want to use V.G. on the side of the 'U' away from the archers.
Try to target your archers onto enemy units which are not quite up to the front of the bridge, to avoid shooting your own men in the back. Some feel that simply blocking off the bridge with spears (not forming a 'U' pocket) is better, since there is less exposure of your men to your own archers. IF you have enough archers to quickly exterminate the enemy before your front line folds(and if you followed my recipe, you do) than this might be a better option for overall reduced friendly casualties.
What typically happens in these bridge battles is that the first men across the bridge get slaughtered in your 'U' pocket. Meanwhile, the rest of the enemy stack gets stuck on the bridge, packed in tight, with their unshielded side facing your archers. And the archers slaughter them, like shooting fish n a barrel. Meanwhile, the enemy troops in the pocket (what is left of them) rout. As the routers flee back across the bridge, they take more casualties, break up the formation of troops trying to get across the bridge to attack your spears, and spread decreased morale to the enemy. IF you have enough archer power (and 8 units of Trebizonds is way MORE than enough) the battle turns into a big turkey shoot, with as-yet-unrouted units standing on the bridge waiting their turn to take an arrow.
I have dozens of heroic victories in RTW and M2TW (on H/H and VH/VH settings), based on this one tactic, alone; many of them with less than a full stack against multiple enemy stacks.