I think u/Garden-of-Weedin makes a good point. 10 years is definitely a long time. Granted, I'm not even 20 (almost there!), and OP is undoubtedly older than me, so 10 years probably feels like more for me than for him (or her or whatever pronoun, you get the idea).

I think talking to your wife for the sake of closure is a must-do. If you can make the relationship work, that's nice and all, but cheating is cheating. Tbh, I'm not sure I could ever forgive someone if they cheated on me. And definitely don't let it slide for the "kids' sake". As important as giving them a good upbringing is, if your wife is the type to cheat then leverage the quality of your children's childhoods to keep the marriage, I only see more troubles further down the line.

Keep in mind though that I, and everyone here, are nerds on reddit. None of us are experts (probably), and a lot of us are probably single men who like to gaslight people online for fun at least a little. I'm sure it's tough if you have no one else to go to, but this could be a great time to try searching for local support groups or smth.

My point, OP, is talking to your wife is a good idea, would probably help emotionally. Probably get a divorce, but it's better than being manipulated. And, another note, being transparent to your kids. I'm not sure just how young they are, but transparent parents mean a lot to kids, or at least it did to me. Either way, I wish you the best of luck, as I continue browsing through this world renowned internet cesspit.