Read "BobbiJo" too! She's doing an excellent job writing about the trials, tribulations, and most importantly, the solutions she comes up with to get her marriage back on track.
However, the reason it's tough for her to find an answer for you is this: I'm sorry, but you can't "work" on your marriage if there is an OM. Especially if she's lying about it and ridiculing you. While this is totally normal gaslighting behavior, it is even more disrespectful than starting with the truth.
Soo, what you say when she asks how to work on your marriage, is that you both must agree that there will be no 3rd party involved while you are giving the marriage the best chance possible for the sake of your daughter. Do not allow her to lie to you or ridicule you. Tell her you are serious about giving your family a chance but it will take honesty and commitment.
Are you acting like the man she fell for 10 years ago? Are you acting like the father you want to be?
What are the reasons she comes up with that makes her consider working on your M? These are the things that are working. Why did she move back home? Could the A be over?