This has already but desperately needs repeating:
This isn't some random dude she talked to, spent time with and kissed. He is her ex boyfriend. She has history with him. She clearly still has feelings for him. The OP shouldn't have to deal, cope or settle for anything. His feelings, time, money and energy are too precious to waste on someone who isn't 100% faithful to him. It's not his job to try and fix it or wait for her to sort her emotional bullshit out, while he sits on the side like a chump.
It's not the kiss. It's the fact she is still in communication and still in contact with her ex who clearly has feelings for her and clearly could give a rat's ass about the OP. Trust has been broken. It would require the ex to disappear for good and for her to sort her feelings out. All the OP has to do is decide whether or not he can trust her to break ties with her ex and move forward, once and for all.
I doubt it. I wouldn't bother. It's not worth the heart ache. She fucked up, not you. So what if you break up. It's her lose. You're young. It's not like you're married, have kids or any serious financial obligations together (right?)You'll be on your way to finding someone who will actually respects you. She's on her fucking own. Just telling it like it is.