Louise, first and foremost, don't take the blame for your husband's bad behavior. That's a choice HE made, and HE made it alone. You didn't 'push' anyone together.
Secondly, try to spend some time reading in the Healing Library - there's tons of articles and information in there that might help you and give you some guidance.
Unfortunately, most betrayeds find out their spouses didn't stop contact with their affair partners once the affair was discovered and they promised to cut all contact. They usually just take the affair a little deeper underground and try to hide it better while waiting waiting for the heat to die down.
Your husband is forcing a very uncomfortable, unhealthy, and insecure dynamic on your entire family. Does he honestly think the kids aren't going to feel the discord and tension in the house? Does he think the damned world revolves around HIM and he has the right to visit this dysfunction on innocent children who have no choice but to live in this hell he's created? Cheaters are some of the most self-entitled, selfish, self-centered people on the planet, so I'm not too surprised by his behavior. But you shouldn't stand for it.
It would probably be well worth your while to consult with an attorney and find out exactly what your rights ARE, should you need this knowledge down the road. Knowledge is power, and it never hurts to know exactly where you stand, Louise.
I'm sorry you find yourself in this position.