My advice/opinion is probably going against the grain but too bad.
You say you love her, so do the right thing. If you love her (truly) then you'll agree that she deserves the best right? You simply can't give her that in my opinion, and you've already had enough chances. I also don't think there is such a thing as "making up" for what you did.
I think that realising (and admitting to yourself) that she is a wonderful woman and mother who deserves a partner who will *never* betray her or her/your children and wishing her well is the best thing you can do I think if you truly love her, you need to want the best thing for her and your family. In my opinion, that doesn't involve you in a husband/romantic sense.
I think trying to be a friend to her and a good dad to your kids is all you can do - and I think that's the fairest thing to do. She's been through enough, mate. Cheers and best wishes .