A Mongol Prince and his Horde convert to Christianity. He is invited to restore the Byzantine Empire. He takes Constantinople, and becomes Emperor himself. As Emperor, he has a powerful, efficient army, and establishes an efficient state to support it. (The Mongols are pretty good at that.) Mongol Byzantium whacks the Moslems, regaining Anatolia, and whacks the Slavs, regaining the Balkans. These territories are incorporated into the efficient state. Meanwhile the Khan-Emperor also holds a great dominion north of the Black Sea.
Venice gets into bad trouble in Italy, and yells for help to its nominal overlord. (Venice was originally a Byzantine vassal; that's why it was never part of the Holy Roman Empire. By this time, that relationship had lapsed, but when the Venetians get really desperate, they call it back up.) The Khan-Emperor shows up and rescues them, and then takes over Italy, demoting the Pope to Patriarch of Rome.
Meanwhile... The battle of Las Navas de Tolosa was a resounding victory for the fanatical Almohad Moslems. All four of the Christian kings present were killed, and the Almohads conquered everything south of the Pyrenees. The Almohads began forced conversion of Iberian Christians to Islam. This resulted in a Crusade by France and England, which hit the Almohads hard, but ended with nearly all the Crusaders killed. A generation later, the Almohads surged into France. This time the Holy Roman Emperor joined the Kings of France and England in the Crusade, but the Moslems defeated and killed all three at Bourges, and pressed on to besiege Paris.
The siege was broken by the arrival of the Khan-Emperor and his army, who routed the Moslems. Another Imperial army was in Spain, cleaning up there. The Khan-Emperor decreed that all the redeemed lands must pledge allegiance to the Empire. He also demanded recognition as the new Holy Roman Emperor (or more precisely, that his position as Emperor included that title).
Over the next three centuries, Hungary, Poland, Lithuania, Novgorod, Scandinavia and the British Isles are all absorbed.
Bingo.