I've been there where you are. You aren't alone.
As far as your questions, I'll try to answer with what worked and didn't work for me.
When is enough...enough?
Enough is enough when you say it is.
How can we reestablish trust if he continues to lie
You can't.
How this would affect our kids.
Either way it's gonna effect the kids. Even if he turns into a saint tomorrow, what's happened still will effect them. Sure, there's ways to minimize it. Keep the conversations out of earshot. Answer questions in age appropriate manner. That's kinda getting the cart before the horse, though.
I just don’t know what to do
If you're asking us if you should get a divorce, we can't answer that for you. What I can say is some of the things I did.
When I couldn't sleep, I planned. Like a mad scientist. Swear it was 8 pages long. I felt that if I wasn't sleeping, that I might as well try to be productive.
When my wife wasn't taking responsibility, I did the 180. It was really hard and against all the habits I had grown accustomed to over the marriage.
Some of the quotes i heard here that helped me the most at that time was
"All the little things you do to show love. The treats from the store. The kiss goodbye in the morning... none of those stopped them from cheating, so they aren't worth all that much, are they?"
"Why did they cheat? Because they wanted to.
Journaling helped me a lot too. It helped me filter the lies, track my thoughts, plan, etc.
The biggest, most important piece of advice that I can give you right now is take care of yourself. Situational depression is all to common in betrayed spouses. Work out, eat right, don't drink to get drunk. Right now, you are demonstrating to your kids how to properly cope with horrible shit when it hits the fan. When you start thinking about doing crazy things, just imagine how you would want them to react if this happened to them after they're grown up.
Again, you aren't alone.