The Horde always take Khazar and keep their main army there when they do. They'll also occasionally attack Georgia, Volga-Bulgaria and Armenia. For me, they once attacked in 1227, with, if I recall correctly, 4000 soldiers in Volga-Bulgaria and 12,000 in Khazar.

I purposely had 1 byzantine infantryman stationed in Volga-Bulgaria at the time. When the Horde appeared, I made him withdraw into the fort. Meanwhile, in Khazar, I almost entirely destroyed the invading army with a defending force of 8000 steppe cavalry, byzantine infantry and a range of peasants.

After I had almost entirely wiped out the invaders in Khazar, I moved my surviving 3500 soldiers from that province into Volga-Bulgaria, where they had taken the castle (that poor, valiant infantryman died...). In the next year I easily destroyed the remaining Horde soldiers.

I defeated them in a matter of three years.