You have a lot of years together, this isn't just some 6 month to 2 year relationship and you have children together and a history of obviously having loved one another.
I'm not sure what the point of this thread is now? You say you don't love him anymore so what is the point of your Original Post? What does it matter that he won't tell you the nature of the affair if you're done with him?
If you are done with him then leave it alone and concentrate on getting over the betrayal and adjusting to a life without him.
If you did decide to stay with him then that is your business and I'd not try to talk you out of it however; I would strongly suggest that you get marriage counceling because I cannot fathom how anyone would be able to learn to trust and forgive without a unbiased third party mediator so that somethng like this wouldn't happen to either of you again because you'd have the communication skills and proper relationship crossing blocking tools in place to nip relationship boundary crossing actions in the bud.
Good luck to you, Sweetest.