You never said distance was an issue in the OP
Then I would try fighting fire with fire and use the Cumans or Novgorod. While unit for unit, they don't match up to the Mongol Elite, you can use horde tactics to drown the Mongol in numbers just as long as you can keep one front. The only Teutonic Order campaign I ever lost was due to both Russias being allied and declaring war on me. While I was kicking their asses at first, eventually I couldn't keep up and they won through attrition
Other than that I'd suggest Poland (Late Game), Hungary (Late Game), Byzantines (Early Game), or Khwarezmian (Early Game)