This thread is a good example of how people who have had affairs and then want to come back find it hard - understandably so - to actually realise how fundamentally their actions have changed things. They always say the same things. 'Massive mistake' 'complete regret' 'winning them back' 'make it up to them'.
To the betrayed person, and indeed to anyone else with a shred of emotional intelligence who think about it for more than a minute, those comments only serve to show how completely they DON'T GET IT, and even perhaps how being able to think like that makes them the kind of person who has an affair in the first place.
Because that's what it's really about - being a certain kind of person. That's it, ultimately. Your affair wasn't a mistake. It's something you made the decision to do because that's the kind of person you are. And that's what your wife now knows - that she thought she was married to Funkywotsits-A: a loving, caring partner, dad, faithful, her rock, etc. etc., and she's now discovered that that person doesn't exist - and she was actually married to a bloke called Funkywotsits-B - who is the kind of man that she'd run a mile from. A cheat, a hideous twat who would look at his own children and turn his back for the sake of having fun sex with some other woman. His own children. Her worst nightmare. Now, you've changed your mind (no, not 'realised your mistake, you've just changed your mind - own it) - and you're asking her to agree to spend the rest of her life with a new husband, a nasty betraying cruel cheat called Funkywotsits-B, who she didn't know existed until he turned up wearing her beloved DH's clothes and ripped her heart out.
That isn't how you see it, because you feel like the same person. You will find it impossible to accept the above, because to you, you're the same lovely loving bloke she married who made a mistake. How can you live with any other interpretation? But the truth is, you always were Funkywotsit-B. She just didn't know it yet, and you don't truly appreciate the difference.
So, to her, and to most people on here, talk of a 'mistake' just shows how little you understand of what's happened. You didn't make a mistake, you simply showed what your character is when the chips are down. Talk of regret and making it up to her forever means nothing - because the important thing is, you are a person who would do this as opposed to a person who wouldn't - and when we're talking parenting, young children, family life, that's not a factor that she can afford to be less than 100% about. Now she knows what your idea of 'better or worse' is really like, now it's been tested, what do you think she should do? You CAN'T 'make it up to her'. It's BEEN DONE - the marriage has been destroyed. You can' 'not be' the kind of person who would have an affair. So, you're asking her to spend the rest of her life in a second or third-rate relationship with a man who, when the chips were down, was willing to piss all over her and his children and walk out. You're asking her to do that, instead of move on and look for a better, unspoiled relationship in which she might have peace of mind, for your sake. The person who wouldn't even stay faithful to her.
So, I don't know where you can go with that. There are people on here who have moved on from affairs. I'm guessing that the reasons are essentially twofold - either they don't care enough about the fact of infidelity (and thus don't feel the way outlined above) and see the marriage in completely different terms, or they make a conscious decision to accept the broken relationship in order to maintain other parts of their married life that are more important to them. Your wife does not seem to have either viewpoint, so essentially you are offering her nothing. Worse, what you are offering is for her to continue to live this hideous hurt and uncertainty forever, rather than move on from it.
I honestly think your best bet is to really try and understand this, appreciate with honesty and good grace that she is being utterly sensible in wanting to end the marriage, and try to build a new friend and co-parent relationship with her instead.