Kick him out tell him you need space to think. Make him face up to what he is losing. Call family and get support and think about what you want to do, how do you want this mess to be resolved? You can't go back to the happy family with the lovely H who doesn't fuck the family friend. You need to think about you and your children and what you want now.
This.
His behaviour since you found out is perhaps even more offensive than him cheating!
He is insufferably self involved and you are enabling that by being passive in the decision making around this.
At least tell him you need a few days / a week to think about this and that he's had her AND you to confide in play the victim to for over a year, so you will be telling a friend / family member what's happened to get some support for YOU while you think about what to do.
None of us can tell you what to do but my god please, please, please leave this man. He's that awful combination of horrible and pathetic.
I'm getting the second ick just hearing about him! Could you really live with him happily again? Sleep with him again? Laugh and joke with him knowing what he's done?
He'll do puppy dog eyes and feign sadness you've asked him to leave but he will be sad about leaving the house and potentially being outed, not about your emotional well-being.
You need time and space away from him to be able to see him for what he is - someone not worth you losing your self respect for.
You won't have broken up the family, he will. And that term doesn't matter because your children will still have a family, their parents just won't live together. It's perfectly possibly to be healthy coparents instead of remaining in a deeply unhealthy relationship.
It's clear from your posts that he is the decision maker in the relationship and dictates not just what you do but how you should feel / are allowed to feel. That's awful and it isn't love. And it isn't something you should want to model to your children, who are learning from you what a relationship looks like.
Ask him to leave so you can have space.
Call a friend and tell them what's happened, talk it through.
Then hopefully you'll have the epiphany that he's an icky wanker who doesn't deserve you.