You wife knows exactly what she needs to do, and God has clearly shown her, the rest is up to her . It does not mean that she will do the right thing, because the right thing is to leave the man now and cut off all contact for good, but you can keep praying that she will. You say that she has nowhere to go, but you have given her a solution to that, so she now has no excuse. Its her decision.
As for thinking of ending a marriage being sin, under certain circumstances God allows divorce, and I do know people (including myself) who God clearly led to separate and divorce their spouses for serious reasons. Adultery is so very very serious, after all had she lived in OT times she would have been stoned to death, and I think that people forget that it breaks the marriage covenant and destroys the trust.
You have decided that you can have her back if she chooses the right way, some cannot, and I don't condemn then at all, because even God knows that for many there is no way back from such serious and appalling betrayal.