Gotta say, you keep going the same things expecting different results. People have told you to detach, do the 180 and you’re wondering how to get respect back? Dude, she is in an active affair, rubs your nose in it repeatedly and you’re worrying about getting respect back? You keep hoping to nice her back, how is that working for you? At some point you’re going to have to face the grim reality that what you are doing isn’t working.
I get you can’t kick her out and INS [censored]. You need to detach as much as possible from this woman and focus on your kids and yourself. As someone else pointed out, treat her like a neighbor. Cordial and polite but that’s all. Make her see what she is losing. Become the best person that you can be, the best father you can be. Don’t let her hoover you back in will fun little trips; all they are are little crumbs to keep you around. They have done nothing to get her to turn back to you and away from the OM.