Man, that's a tough one. You need to have compassion. But on the other hand, you cannot "fix" her unless she is willing to fix herself, and to the extent she is willing to further damage herself and potentially damage your own family/household, you aren't showing love and compassion by allowing her to do it under your roof. In fact, you are probably indirectly enabling it to an extent. But the other consideration that makes it even more complicated is the kids. Man, that's a tough one. I'm going to throw out a couple of things, but this is by no means any sort of "expert" advice or anything.
First off, in terms of her, I guess I would probably sit down and have a "final" talk. In that, I would make it clear in no uncertain terms that you love her and want to help and offer support. But I would also make it clear that she has long since exhausted any sort of "second chance" she might have, and she can stay IF AND ONLY IF she agrees to (1) attend and complete a rehab program, and (2) join AA and faithfully attend ALL meetings. Her AA sponsor is then the one in charge or managing her sobriety, which takes the onus off of you and your family and puts it on someone who is qualified AND is a detached third party. It also makes her accountable. And if she cannot agree to both of those conditions AND keep those two commitments, she is out. And I think you can probably get some good advice from her rehab counselor(s) and/or sponsor about your own role in all of this, and how much grace to extend if she slips up and falls off the wagon. They deal with those situations, so they know what generally works and what doesn't. The thing is, with an addictive personality, the odds are that even if she tries hard and is committed, she WILL at some point fall off the wagon. That is probably understandable, and in some situations, you can probably work through it if her counselor/sponsor agrees that it is a temporary setback. But they would probably be in a better position to advise you in that regard. But if she won't or says she "can't" do both of those, I guess I would feel that it is a deal breaker because you simply CANNOT help her through such a destructive addiction problem otherwise. I would let her know that. Tell her those conditions are nonnegotiable, and they are that way because you care and that is what she and the kids need, and it is what YOU need. And if she loves her kids and loves you, she will need to understand that.
Second, have you thought through what happens to her kids if something happens to her? It sounds like their father is still in the picture, in which case he would get them. If so, whether that is good for them, bad, or indifferent doesn't really matter. There isn't much wiggle room and that isn't your issue. But if he isn't in the picture, then what? Would you and your wife be willing to take them in? If so, you probably need to have that discussion with your sister at some point. I'm not sure what that point is. But you and your wife need to be on the same page beforehand.
Man, either way, it's a tough situation, and I feel for you. Hang in there. If it means anything to you, you are all in my prayers.