Whether the rest of you agree or disagree with Supermannnnnn in MY opinion he's right, walk away there's nothing more to be said.
Facts:
• She wanted to break up, uses her kids as the excuse. Maybe rightfully so…
• A week later she's dating someone else. HELLO… a week later!?
• Then wants to get back together yet to flake out once again saying, "wait". Wait for what!?
Sorry but you owe her nothing. I'd tend to believe she was already eyeing up this other guy and keeping her options open, a.k.a. backup plan. When she asked you to 'wait' in my opinion it was for her own selfishness to evaluate his potential over yours. You handled that well and gave a straight forward, “NO”. That's it, you're done.
Dude seriously… there is no need to go back and express that, “Ah… yeah…. I'm not going to be talking to you anymore because you wanted to compare the new guy's worthiness to mine. So I'll see you, OK?”
A REAL man or woman sees someone pulling bull$hit like this and walks away from it. Come on people… seriously…