InterimRent,
You say you "really want to try and work this out." The sad truth is that, at the moment, what you want doesn't matter because she is in control. Until she wants to try to save the relationship, nothing good is going to happen to it. This illustrates one of the most important principles of human relationships -- the one who values a relationship the least determines its future.
What can you do about this? There are basically four options:
1) Keep doing what you have been doing. Beg her to stay and tell her you'll do anything she wants if she'll stay. Keep hiding the truth from your kids and do everything else you can to protect her from the consequences of her actions, in the hope that this will convince her to stay.
2) Don't press her to return but try to be the best H you can be (be especially attentive and loving, offer to do things for her, try to get her to date you and spend time with you), so that she will eventually see what a mistake she is making and want to return to you. Distract yourself with taking care of the kids when she is gone and with various interests and activities, in order to try and maintain your sanity. If you have proof of an A, expose it to everyone she cares about and to her AP's W, in order to break it up. Keep this up until either she comes back to you or you can't take it any more.
3) Do the 180 until she loses interest in her AP (assuming she's cheating) and decides to return to you. If you have proof of an A, expose it.
4) Regain you dignity and demonstrate your self-respect by telling her that, if she wants to leave, she should leave because you don't want to be around someone who doesn't love and respect you. Give her a very short window of opportunity to return (like a few days, at most) and, if she doesn't, immediately file for D. Start to rebuild your life without her by focusing on your and your children's well-being. Meanwhile have as little to do with her as possible, limiting your conversations to things relating to your children and the D.
I think you're beginning to figure out that Option 1 rarely if ever works because most woman don't want to be married to a man with no self-respect.
Option 2 is what the call Plan A over at Marriage Builders. It sometimes works, according to them, but it takes months and sometimes years, and it usually takes quite a toll on the BH's physical and emotional well-being and self-esteem.
Option 3 is what they call Plan B over at Marriage Builders. The idea is that, if Plan A fails, go to Plan B to preserve your sanity. Of course, in the rare cases in which does work, it means the BH is the real Plan B.
Option 4 is based on the principle that an intelligent, strong and successful man should not waste his time with a woman who does not appreciate him. It also just happens to be the most effective method of winning back a WW's respect and interest. It doesn't always work, but it works a lot more often than either Option 1, 2 or 3. It also has the advantage that, when it doesn't work, it prepares the BH for a new and happy life with the WW.
If your like most BH's, you've become so codependent that you can't imagine living without your WW. The sooner you start acting like someone who can live without her, the sooner you will find out if there is anything to save in your relationship and the sooner you will start to heal.
While you are thinking about all this and deciding what to do, I highly recommend that you read "No More Mister Nice Guy!" by Dr. Robert Glover (you can download it for free here: https://7chan.org/lit/src/Robert_Glover_-_No_More_Mr_Nice_Guy.pdf). It has helped innumerable guys like you overcome their fear of separation from their WWs, regain their emotional independence and lead happier lives.
Good luck.
[This message edited by Alchemy at 4:26 PM, May 1st (Monday)]