Welcome to SR. Your in the right place to begin your recovery, We will help you through this. But the first thing you must know are the three C's. You didn't cause it, You cant control it, and you cant cure it.
The Addiction is one thing and the Affair is another, but both tell me that he is only thinking of himself. He has what I call the "ME" syndrome, it's all about me {him} and only me {him}, nothing else matters.
The only thing you can do is protect you and your children. When he is ready he will come around, but only he knows when he hits his bottom. Nothing you do or say will change anything.
I am sorry for your situation. Keep reading, keep posting and take care of you and your kids. He must seek his own help you are too close to him to help. He must concur this alone you can't help him.
Sending strength and prayers.