It sounds like you are the only person who is confused.

Your wife was cheating on you and has announced to the entire world that she is done with the marriage. And as an outsider it seems like you two would be better off as sex-partners and roommates than you would being committed to each other. And you even wrote that you wo are faking this for your kids.

Personally, I think you need to be a realist. I would look into a non-contested divorce with a written agreement to cohabitate for 3 years pending some major change to your current situation.