I list important civs and their most important rulers as well as their capitals. The most important civs to the Mongols are in bold. I'll let you decide what others go in.
Mongols-Genghis Khan Temujin, Karakorum
Sung (Chinese)-Emperor Tu-Tsung, Hang-chou
Chin (Jurchen)-Emperor Wei Chao Wang, Peking
Hsi-Hsia (Tangut)-King Li-An'quan, Ning-hsia
Koryo (Korean)-King Ko-jong, Kaesong
Tibet-Lama Phagpa, Lhasa
Japan-Shikken Hojo Tokimune, Kamakura
Nan-Chao (Thai)-King Tuan Hsing-Chih, T'ai-Li
Annam (Annamese)-King Thanh-tong, Hanoi
Pagan (Burmese)-King Narathihapate, Pagan
Kara Khitai (Khitan)-Khan Kuchlug, Balasaghun
Cumans (Cuman)-Khan Kotian
Azerbaijan-Atabeg Ozbeg, Maragha
Khwarazm (Iranian)-Sultan Muhammad II, Samarkand
Delhi Sultanate (Turkic)-Sultan Iltutmish, Delhi
Abbasid Caliphate (Arabic)-Caliph Mustasim, Baghdad
Egypt-Sultan Baybars I, Cairo
Seljuks of Rum (Turkish)-Sultan Kay Khusraw II, Konya
Georgia-King Giorgi III Lacha, Tiflis
Bulgars-King Gabdula Chelbir, Bulgar
Byzantines-Emperor Michael VIII Palaeologus, Nicaea
Russia-Prince Mstislav, Kiev
Hungary-King Bela IV, Buda
Poland-King Boleslaw IV, Cracow
Germany-Emperor Frederick II