Now that you've explained all that, I agree that there's no point in having any contact at all with OW and her H. He's obviously putting his head in the sand - and as for what does she have to gain, well it's pretty obvious.
She sounds like a narcissistic woman who likes the idea of having more than one man being stupid around her. She needs to lie to her H and you, to hang on to her marriage, which she clearly doesn't want out of. This affair was probably just an ego boost for her - and not some big love affair, although I bet she pretended to your H that he was her "soulmate" and this is why he can't let go. Sooner or later, it might occur to him that she was just playing him.
The thing is, him contacting her gives away the lie to his feelings, doesn't it? If this had meant nothing - and she had meant nothing, he would have been able to break the contact easily, especially when he had so much to lose. Regardless of him staying in contact with her, what's more serious perhaps is that he has lied to you throughout about the extent of this affair.
You really must stand firm and not give into any emotional blackmail or tears. As I said before, all his tears will be for himself and not what he has done to you and the DCs. If you relent now, any respect he has for you will be completely lost.
In this situation, your best plan is to act as though you fully intend divorcing him. Get some legal advice (you don't have to follow through) and only have conversations with him about the DCs, nothing else. Work out how you will manage as a single parent, get the finances sorted. Knowing you can cope logistically and financially will help enormously. Emotionally, it will be much harder to detach, but you must for your own sake. Fake it to make it, because if you show to him that you are resolute, will never take him back and intend getting on with your life and meeting new people, eventually you will actually want to do these things.
If your H really has been a stupid prick who made a never-to-repeated mistake; an intrinsically good man who got carried away, then he needs to realise what he lost for that mistake and fight like mad to prove he is worthy of you. But given what's happened and that he still felt entitled to deceive you even after discovery, this should take months or years to win you back, not easy words and tears and a few weeks on his parents' couch. He needs to live on his own and have the children at his new place - living the life of a single parent.
If you insist on this, it's a win-win situation for you. You can watch his actions and over time, decide whether he's worthy of a second chance, or you might start to see that actually, this behaviour was just the tip of the iceberg and you are better off without him. That he's not an inherently good man after all. You will only be able to make this decision from a distance though. If you let him back imminently, you will try to convince yourself that he is a good man and you did the right thing - and you will lose all your objectivity, as well as any chance of a good marriage with a partner who respects you.