Ok, I've just fought my two first battles against the Horde in the same campaign. Luckily, they only appeared in Khazar, so I fought them in Georgia where the terrain was to my advantage.

The first time they invaded with some 5,000 men against my 3,500 (for some reason they didn't attack me until 1233, so I had a few years to send troops from all of my empire).

It might be a good idea to just point out that I've modded the Byz unit roster to their advantage (I prefer seeing a powerful Byzantine Empire than a powerful Fatamid Caliphate). The Byzantines can now recruit Order Foot, which have the same stats and building requirements as Italian Infantry (I originally intended to make a whole new unit like in the Pocket Mod, but found it too complicated and looked up a unit that wasn't already used by some faction), since it would be realistic for them to have superior spearmen to European factions. I've also turned the Latin Auxillaries into good polearm units - like Halberdiers but with somewhat better stats and less armour, still only recruitable in 4 provinces and cost more than Halberdiers though. This usually means the Byzantines fare a lot better when the AI plays them, and makes my own Byz campaigns a walk in the park (I hardly ever play the Byz and only keep them as a worthy opponent though).

Anyway, my Jedi Emperor was on a hill with the traditional "Arbalesters behind spearwall" tactic and with Latin Auxillaries on the flanks. The battle was looong (and rather dull) and ended with 1300 dead Mongols and 500 dead Byzantines. I also managed to pick off their general, a 4-star one which, dissapointingly enough, was the best general the Mongols had. The next year they returned with 6,500 men under a 2-star general. This time I had an amazing terrain advantage in that I, err... it's a lot easier (and funnier) to just draw it up:

The brown is height curves, I guess you can tell where the impassable cliff upon which my army stood was. Green is Arbalesters, Blue is "OrderFoot" (let's call them Hoplitai, like in MedMod, they have the same stats as them) and LA's, Black is the initial position of my cavalry, Red is the later position of my cavalry as more arrived with the reinforcements.
For this battle, I had way too few Arbs, so I ended up with three Katatanks on my right flank, and 4-5 (depending on when) Pronoiai to my left. It was horrible. Initially I had great success in mowing down the general, several units of MHC and MW's, but then reinforcements with endless amounts of MHA's arrived. I used my cavalry to slay the odd MHC and MW that still showed up, but mostly it was sitting there, waiting for time to pass. When I finally got some BC's from my reinforcements, I first used up their arrows and then tried to chase the MHA's off with little success.

In the end the Mongols lost 3,100 men (I killed all the prisoners) and I lost 600. A pretty good casualty rate one would think, if one didn't consider what a huge advantage I had in terrain. 95% of my casualties in both battles were my Hoplitai who could only stand there while being peppered, and I doubt I would have done better if I had had Halberdiers. I recall my Polish campaign in the Pics & History-thread, where I experienced similar situations. They just keep calling in wave after wave of untouchable units. I think there were 10 units in each wave, and I couldn't even count the waves. Next time I'm bringing at least 4x8 units of Arbalesters, to counter the 16x10 enemy MHA's.