OP you have given us way little information. How was your parents marriage before this blow up? Were you not close with your mother before? You mention you are close with your father but nothing about mother. Also were you upset or angry by the actions of your father? From the information you have shared (that you are staying with your dad and his mistress), it seems that you are not bothered by what is happening.
Now based on the limited information you have given, your father is clearly the guilty party here. He did something that is morally wrong. Hopefully you understand that. Your mom is very hurt and must feeling betrayed (slightly even by you because you chose to stay with your dad and his mistress). In that emotional state, she is making a unreasonable request to you (to go no contact with your dad). If you want a good relationship with mom, you will have sit down with her and reach an understanding and compromise with her. Maybe you can go to therapy with her like some other commenters suggested. One thing I am sure about is that if you side with father on everything, you will end up estranged from your mom.
Well if the that is case, there is no reason to compromise to have a relationship with your mom. Being with your dad is the way better option for you (since he loves you and you love him) and maybe it is the best option for everybody involved.
Still I hope you understand that at some level what your dad did was morally wrong. Irrespective how crappy things were, the correct order would have been to seek divorce before hooking up/dating somebody else. Learn from the mistake he made.