If you are going to let her stay, you can't snoop and you can't ask about the relationship. It's incredibly unfair, but she needs to get there on her own. If you apply any pressure, she'll never notice your 180's and not applying pressure is usually a 180 in itself.
I get wanting her to move out if the affair continues. My ex and I were in the same place. She "ended" it when I found out and 3 weeks later I found that they were still talking. I told her that I couldn't live with that and she had to leave or stop all communication. She's been gone for 6 weeks and we barely speak.
Having her move out is a risk. However in my situation, I'm glad I did. I feel better about myself and the stress levels in the house have almost disappeared. She was difficult to live with over the last few months and the kids and I are closer than ever, I've lost weight, started reconnecting with friends, etc. Just know that having her leave could mean the end of everything. If it is something that you need to do for yourself, to heal, then do it. Just give yourself time to be sure. You can always get her to move out, it's very hard to get her to move back in.
M: 38 H: 39 D: 20 S: 18 S: 16 T: 14 Y ILYBIDKIILWY/Affair 01/12/2013 Came Back 01/15/2013 Left Again 02/13/2013