Thanks for the help everyone, good tips all around. Though I have a question regarding one of your tips:

Some of you advise sheltering in my castles and fortresses. From my experience, the AI doesn't storm settlements, it just besieges them. I've been afraid to take shelter in my settlements because I fear being starved out, which usually results in losing a massive ammount of men without killing any of the enemy. Do the Mongols actually attack?