I'm sorry that you ran into trouble with this! Are you looking for fonts to add to a website or fonts to sync to your computer and use in your desktop programs (like InDesign or Illustrator)?
I want to find some fonts supporting Mongolian language. Do the typekit include such fonts?
Typekit does not currently have any fonts which support the Mongolian language. There are a number of fonts available for sync use for Chinese, though:
(The separate filters are for Simplified vs Traditional.)
Font sync is slightly different than downloading & installing fonts, as the synced fonts are not permanently installed on your computer. You will be able to use them in all of your desktop applications, however, as long as you are subscribed to Typekit.
This tutorial shows how font sync works:
I hope that this helps; let me know if you have any other questions.