Missmee, I'm so sorry you're living through this nightmare this thoughtless man has so cruelly done to you and your children. No matter how we may feel about a spouse, or an OW, good or bad, the children should always be the most important thing to any adult. That this man will not do what is in the best interest of his kids, says everything that can be said about him to me. We may want to have sex with someone else, we may want to love someone else, but our obligations are always to our minor children - they need us for practical and moral support. There is no excuse for his behavior.
You're stuck in limbo right now, it sounds like because you don't know what to do with him. It sounds like you're tried appealing to his better nature, his sense of responsibility, his morals, your long relationship - whatever - and nothing makes any difference because this is what he wants right now. He wants this other relationship. There is nothing you can do about that. The "best" case scenario is that he tries to have BOTH and tries to force you into a never-ending pick me dance, which you should never do. Don't do the pick me dance, it's degrading and it doesn't work and it only gives HIM more power.
All I think you can really do in this situation is tell him to get the hell out and go see a lawyer and file for divorce. Do whatever you can to make it hurt - the love is already gone or he wouldn't be doing this. Get your power back through action, stop trying to work with him or change him. THIS BEHAVIOR IS A DECISION ON HIS PART, IT IS HIS CHOICE, no matter what we or any decent people think of it. It's not a decent thing to do, but it's what he wants. Other than trying to have you both which is....degrading and unworkable and a gross insult to you and your children.
Go for the divorce. You don't have to go all the way through, you can pull it at any point. I personally would go all the way. Get individual counseling for yourself, for the support. Take good care of yourself physically and mentally. Get whatever help and support you can get from family and friends and don't hesitate to tell them why. Don't rant about it, people can only take so much (I'm a ranter so I know), but be honest with everyone, let them know what you're going through. Filing for divorce will give you back power in the situation.
Personally as you say this woman is living at home with her parents, I would let her parents know what she is doing. You can do this through a letter, or email, or whatever. You might want to check with an attorney first about the wording but I think just letting them know what she's involved with might help to break it up from that side. I would not rely on getting this man back, frankly, because at this point, I would only think of him as whatever financially he could provide for the kids. I wouldn't want him back after this treatment. Maybe you shouldn't either.
Take care of yourself like you are a Queen. Because you are the Queen of that household and that family and you need to be as healthy as you can be for yourself and them. I think if you kick him to the curb, you might start feeling better about a lot of things. The worst part of any cheating relationship is the LIMBO part, to me. Stay with us and let us know how you're doing, and just also realize that while we're all rooting for you, not all of us, including me, know what is exactly right for you at this time. We can only say what we think or know and you can pick and choose, but believe me, we all do care.