WHY DO WE DO IT?

[CD Bear]

I really don’t have any idea how to do this. She still talks to, emails, texts him all the time and admits it, so I don’t really know how to make sure she is not talking to him while here. Any suggestions?

Transparency requires that she has no passwords on nanything and you are free to "have a look" at your request. If they are just friends, "have a look to confirm" Trust but verify.
Your house; your rules.
Your boundary.

Honestly, my confidence is soaring. I have really and truly dropped the rope and am just getting along with my life and my kids. The tone of the last post was more anger at myself for even agreeing to it and at her for even putting me in this situation knowing I am a “nice guy” and would agree.

Simply awesome, my friend!! I find it ebbs and surges a little but considering I opnly confirmed the affair 5 weeks ago, I feel great.
Go buy No More Mr Nice Guy and Hold onto Your N.U.T.'s.
Very good books, IMO.

I am honestly not sure it IS a positive, but also know there is NO WAY I could live with myself if I refused to let her stay and she wound up on the street or living with some unsavory person and risk exposing my kids to them.

Stick to your personal values on this. You value your kids first; if this requires some uncomfortableness to house you W, so be it. Priorities.
You could consider changing the custody. "Until you find your own quarters, I will keep the kids her" Just a thought if she doesn't abide by the boundary.

she is not interested in that since she “can’t forgive me for being cold” and doesn’t think I can ever forgive her… I told her that last one is for ME to decide and not her.

Great response to the "telling you how you feel". As far as the "being cold", I can't recall all the details of your sitch so could you refresh me on that? Did you pursue her feelings on this issue any further? e.g. "In what way did you feel I was cold?"

PS If you were cold, she wouldn't be in your house......