Genghis and his descendants certainly conquered a huge area, but the empire quickly broke apart after Möngke (Genghis' successor) died, into the Ilkhanate, Yuan dynasty, Chagatai Khanate, and Golden Horde. Most of these didn't last more than a century, but more relevantly, the Mongols weren't particularly interested in spreading their culture and largely kept their language a private matter. Subsequently Mongolian (the language) basically doesn't exist outside of Mongolia/Inner Mongolia (China) and culturally there is far less influence as compared to say, the Russian or Spanish empires.

https://en.wikipedia.org/wiki/Division_of_the_Mongol_Empire