I can speak to the linguistic legacy. There many words in contemporary Russian of Mongol extraction, some very central - for example, the word for money "деньги" is a Mongol word, as is the word for horse "лошадь", unsurprisingly. I believe the Russian word for name, "имя" is also from Mongol extraction but I can't find a source for that offhand.

Note that I assume by Mongolian empire you mean the Golden Horde, which included both Mongolian and Turkic tribes.