I don't know how your feelings are on joint-custody, or having her come visit the kids, but you need to be very careful if you don't want her to be a part of their lives.
Firstly, for the love of everything good and decent in this world- and more importantly, your kids- DO NOT get back with the cheater. I'd suggest trying to shrug her off as much as possible, in fact. The kids are yours now, regardless of biological connections- parents are not made by blood, they're made by love and caring. If she pawned the kids off on you every time she couldn't handle them, she's not fit to keep them. Nothing is gained by entering into a bad relationship just so the kids have a mom... because then you're just creating an angry, unhappy household.
I don't know if you plan on sharing custody, or what role at all the mother will play in their lives in the future, but if you don't plan on her being there for them, and want to keep her away from them for fear of more problems, then there is more I'd like to add.
Also, you should start a log book. It can just be a spiral-bound notebook, doesn't matter. What you need to do is log every instance of the mother contacting you, for whatever reason. Example- she calls to ask how the kids are, you record the date and time, and that she called to ask how the kids are. If she calls and says she's broke, wants some money, but doesn't mention the kids, write that down as well. You have to do this for the security of your legal guardianship of the kids. If you don't, and you have to go to court for custody down the road, you will have no record of her disregard for her kids (if that's how it plays out- don't assume "that would never happen"). If, however, you have a record of every time she's called you, and what she called for, she won't have a leg to stand on. My uncle had to do this very thing to keep his drug-abusing, alcoholic, dangeroulsy neglectful and borderline-abusive ex wife (and mother of their kid) from having partial custody- and he's the biological father. You wouldn't believe what he's been through to keep his son, even though he's squeeky clean, has a great job, and everything in the world to back up what a great dad he is, and what a good home he provides. The courts still have bizarre preferences for the mother.
Ok now relax again... that's not to scare you, it's just information. There are tons of things a parent does to protect their kids from the dangers of the world- this is just one of them, so be cool.

(btw i'm not trying to suggest the mother has no rights, or that the mother should automatically be excluded from their lives, but I'm getting the feeling that the mom is a flake, and shouldn't be allowed to have these kids back. I'm working under that assumption.)
Also, I agree completely with the others who have said to never talk badly about the mother. It will only cause problems, and it's just not necessary. Just be a good dad (I know you will be) and let
her actions speak for themselves.