The thing is you can’t prove yourself to someone without getting your foot in the door. You can’t get your foot in the door unless she open the door even a crack. You stated your case to her that you made a mistake and that’s all you can do. If she has even an inkling of hope or interest in you she will reach out again. That’s when you set a date. If she declines you move on again. Every single girl I have ever dated and have broken up with not matter whose idea has reached out again accept for one.