I think you (and we here) should stop beating yourself up. Everybody makes mistakes and while this is a big one, you are doing all you know to do. And that is the problem. To quote a friend of mine, you are like "A monkey working a math problem."
Lets take a look-
First, 6 kids? Its a fair bet you, as a couple, haven't had or made much time for the relationship for a good while. Her role in the family probably took her into the motherhood thing in ways men can't even understand. And that role probably took her away from you and your needs too. So when this opportunity came up, well... and that is an important part - you didn't go looking.
So what has she taken on as her role and where do/did you fit in? Are you a cornerstone or a support person? Remember,not your opinion but hers. Is she an insecure person? I bet so. Maybe we should take a look at her viewpoint? The unthinkable has happened (and with someone she called friend) can she trust her judgments? What can she do to recover herself with you desperate to talk and resolve this situation? How can she do much of anything? Except lock herself up and hide. And we shouldn't forget any hormone changes, like pre-menopause. They can really make things difficult for her and you.
Seems to me she probably wants you out but can't financially afford for you to leave. Loose the house, etc. So I'm thinking you should be the man and move out. Give her a chance to recover herself a bit. And tell her what you are doing by giving her space to recover. Obviously you will keep supporting her and the kids, giving her time to decide what she needs. Continue to try to get her counseling. You should start immediately working with a therapist to show you are suffering too.
This is going to take a while, and if you are serious she will see you are doing what it takes. Sadly, it might not be enough. The issue is hers to decide and obviously she might decide to break up. Hopefully she will see your pain from being without her and the kids and maybe she will see it for what it is - a mistake. Not the betrayal as she takes it. You never intended that, did you?
Its a roll of the dice. However, I'm betting you were willing to do what it took to get her in the first place, and now you are going to have to show determination to get her back. If she is a blamer, be prepared to take the blame for everything that is wrong in her world.
Can the relationship survive? Again, its up to her. The good news? To quote Spock from Star Trek, "...when you eliminate the impossible, whatever left, however unlikely, is possible."
It's going to take years of work and I'm betting you can and will do it.
Good luck!