I can only tell you my experience. When I got into recovery, I was consumed with guilt/remorse/shame of what I had done and what I had put my family through.
Thinking back, I think I actually took quite a while to say the words "I am so sorry". What I did do, though, was work recovery..within days I was looking for another job, helping my dad on his job, GOT a job and paid him back for everything he'd ever spent money on me (he didn't enable me, exactly, but he did help me out).
He and my stepmom noticed the changes. Dad doesn't understand anything about addiction or recovery. Stepmom is an addict. I became a responsible person, and it showed. It was some time before we actually talked about it, and I did say I'm sorry, but I've learned that actions speak way louder than words and he really didn't need to hear that...he already knew. It's become something similar to a chapter in the book of our lives that is closed.
I don't have kids, but my actions did affect my niece (who dad/stepmom raised) and though she is heading down a similar path, she often comments on how far I've come. I can't change her path, but I can be the best role model she has. Her bio-dad is also an A and we've often talked about how I "got it" and he doesn't.
You working your recovery, despite what the ex does, is going to help you and your children tremendously.
Hugs and prayers,
Amy