Re: I'm an idiot. I lost my soul mate.
Sorry you're here.
Don't be the man she deserves. Be the man you want to be. Be the man that others wish they could be.
2X4 warning - you really ****ed up. You should not have had the affair - that was (obviously) a truly poor choice. The perceived pain you have at the thought of another man having your wife is what she has to deal with on a daily basis, but sadly it's not just conjecture, it's reality for her. It is a horrible cross to bear, being a betrayed spouse.
I would agree with laststraw regarding the 'If you love something..." statement. IMO, it appears your current marriage is in severe peril, and may not be salvageable. I would strongly recommend you continue to work on yourself and be the best person you can become.
You state you want her to be happy. Pause a moment and consider what will make her happy, and while I hate to say it, it appears a divorce is that very thing. Yes - you don't want her to be with another guy, and while this is understandable, it is not something you can control in any way. You can try, but ultimately, that's her choice.
If marriage counseling isn't in the cards due to her non-participatory stance, I would truly look at options of ending the marriage to allow both of you to heal, grow, and find another person who may be able to be a life partner. I won't say you two won't get back together, but I will say the probability is slim.
You don't want to hear this, but you've really got to take a hard look at yourself and see what baggage and trouble you caused in the marriage, NOT the affair - you need to look at who you were, and you need to be a brutalist with honesty here. It's between you and yourself. You need to see your failings within the relationship if you want to better yourself for another (or your wife), and work hard on yourself to recover and be a 'better version of yourself.'
I believe you should cut bait, as it were, and find a way to be a better man for the next woman you want to engage after you heal from this matter. I'm so sorry that children are involved. Advice here is also meant to be the best father figure you can be as well.
Here's a different view of marriage (or relationship) to consider:
You are two independent solitudes who choose to be together. Hell - you said vows to each other. Then realize that each of you has your own likes/dislikes/opinions/priorities etc. that need to be honored and respected. BUT! Neither one of you can control or change the other, you can only control and change yourself. If you can grasp that, then you are on your way. So many other positive dominos fall when this perspective is taken.