Let me add ... there is value in you trying to establish communication, for the impact on the kids ... whether you're married or cohabiting or separated, kids' development is impacted by the relationship between the parents. By kids' development I mean emotional resilience, intellectual achievement, IQ, and other measures ...
Sorry bro, that's probably not at all what you want to hear right now. But I'd be negligent in my duty of care as a friend if I didn't share what I've learned.
The good news is that sexual infidelity (perhaps surprisingly) is NOT one of the proven causes of divorce, not even close. That means there is a chance to work things out. I'm not even saying you necessarily need to avoid divorce, if that's right for you. If "healthy relationship between parents" means you live separate lives, that is likely to be better than any arrangement that has you living together or apart and sharing anger / blame / criticism / defensiveness. You have some alternative choices to navigate ...
Of course the bad news corollary is that if cheating is not a predictor of divorce, why is there an issue between you? And clearly there is one ... which means that divorcing won't solve the problem, just bury it. Resolving it would require you both to face difficult questions, and to support each other to resolve the tensions that are getting in the way.
How to do it? Beyond me, except to suggest that a) at this time most professionals seem to be saying the victim of infidelity needs to set the timetable for recovery, and b) joint counselling is the recommended best practice.
If you're interested there are a lot of materials you can draw on. One warning: a lot of the books out there are based on no better than the subjective opinions of a well-meaning individual who wrote about his beliefs. That can be very encouraging stuff to read ... but ... you also have to know that the failure rate in counselling is horrific, with around 65% of interventions failing and half of the successful cases relapsing in the long term. And that's for the intervention techniques that have been formally tested ... the majority of intervention approaches have no scientific rigor behind them whatsoever.
Again some good news: Recently (in the past couple of decades) there has been some real evidence-based science published, which has debunked a lot of myths about marriages and what causes them to break up. That means if you get the best information you can do more things that work and avoid wasted effort on trying things that actually don't work or(in some cases) even make things worse. Very good news if you're serious about results. If you've read this far and get what I'm saying, look for John Gottman online, buy his books, and start studying. You have nothing to lose, and potentially a lot of good to gain, either for your kids, for your family, or at least for a future relationship you can build. You might even be able to avoid being the rather uncomfortable statistic that second marriages fail about 50% faster than first marriages do ...
lots of unsolicited advice there - take it or leave it as you like