Most Helpful Girl
If she's into another guy already and asked you to meet up with her for lunch, I would say that's a no go (in my opinion at least). You don't need to apologize to her. She obviously doesn't appreciate your apologies and I don't even know why she wants you to in the first place. If anything... she needs to apologize to you for waisting your time while she's acting like that towards you.
Most Helpful Guy
I'm confused as to why you would even talk to an ex, especially one that broke up with you. You've apologized? Wrong move.
The only thing to do now is completely cease all contact. Let her show up to the lunch or whatever, but you don't show up. Ignore her texts, ignore her calls. Better yet, block her number.
There's no reason to talk to an ex.