If she's been doing this openly for more than two months, then I'm not sure. You can (and should) still enforce what boundaries you have WITH YOU, and IN YOUR MARITAL HOME, and of course you should still discuss with her what is healthy for the kids (no family therapist worth their salt would EVER recommend bringing an OM around the kids for at LEAST a year or more!).
Although my wife respected most of my core boundaries (no calling or texting OM in front of the kids, no calling or texting OM from inside our marital home, no spending family assets on her affair, etc.), there were others she did not. And there were certainly things like me saying "I have decided that I'm no longer willing to pay for your cellphone, for you to use to conduct your affair," and she went out and got her own and paid for it herself within two hours. So did that accomplish anything? I would contend YES, because a) it was authentic to ME and make me feel better about MYSELF and how I was LOOKING OUT FOR OUR FAMILY, and b) it demonstrated to her that I had firm boundaries that she'd better not cross. She hated me for them at the TIME, but she told me later, when we reconciled, that she respected me for them and that it was attractive!
You're mainly asking how you should ACT around her, and I'm trying to suggest that the whole "unconditional standing, be-their-friend" supplication thing doesn't work.
