I have now played this three times on hard difficulty and I have gotten the "Cuman Flush" achievement where you defeat the Tatars without advancing to the Castle Age, and I found it easiest to defeat the Tatars while you are still Feudal Age. Being in the Feudal Age and knowing of the oncoming rush of Cavalry Archers and Keshiks is coming, you simply work much faster. The last time I played it the Mongols were advancing on my first castle and were doing damage. I had already destroyed one of the Tatar castles and a TC, but needed to finish them off. Subotai et. co. came at that moment and I was unable to mount another offensive against the Tatars for bit, but eventually separated my forces. I headed with one to finish off the Tatars, and the other to hold the Mongols off from destroying my town. I took down the Tatars and that was that. I kept Kotyan Khan in my second castle near the Tatar base for safe keeping.
The first two times I played this I found it direct and just went straight at the Tatars. I think I got a little lazy this time around. Anyway, Pikemen and Elite Skirmishers, a few Knights, Light Cavalry. It is a good mix against both the Mongols and Tatars. Definitely an engaging level, and can become challenging. I didn't see at first why people were saying this was such a difficult mission until my third time through.