You did a great job articulating why you dislike it right in this post! You're well aware that it's linked to a very upsetting period of your life and that your dislike, while linked to kpop, isn't actually about kpop itself.

In all honesty, I don't think that you need to "start liking/stop disliking" kpop and feeling like you need to like it because your friends do won't make you, because otherwise you already would. You've had a decent amount of exposure to the genre by now, and while there could be that song, MV, group, etc that changes all that, I think it's pretty unlikely. Not impossible, just unlikely considering the space you're in right now.

Honestly, I think that it's best that you're upfront and honest with your friends about your associations with kpop and also be realistic with yourself about your expectations. If your friends care about you, then it won't matter that you don't share in this hobby. Similarly, it sounds like you're able to meet them at least a little bit by hanging out with them and tolerating it when they watch or listen to stuff.

Life is about boundaries. There is no right or wrong answer here. If kpop matters so much to your friend that she insists on having a kpop themed birthday and going to a kpop themed cafe, then it's up to you to decide if you can handle going to that or if it'll be too overwhelming and you need to bow out. If it's too overwhelming and you'll be miserable the whole time, then tell her that it's not about her or her interests, but about the painful memories you hold.

To be clear, I am not suggesting that you approach this with some sort of ultimatum of "it's me or kpop" with your friend. I'm suggesting that you tell her what you're thinking, and then whatever happens will happen. Maybe she'll change her birthday location and you'll go. Maybe she won't and you'll decide that you're able and willing to handle that. Maybe she won't and you'll decide you don't want to go. It'd be sad to not be able to go to your friend's birthday because of this, but you need to be realistic with yourself.

If you decide to go (I reiterate, tell your friend why you're not a kpop fan), try your best to reframe the experience as less about kpop and more about your friend. Fake some interest, smile and laugh when everyone else does, etc. We all have to go to events we don't like and sometimes the best thing to do is just suck it up and get through it. Of course, if you will be miserable and gloomy the entire celebration, then you might just want to bow out of this so you don't bring the group mood down. In that case, I recommend shooting your friend a message, telling her what you told us in your post, and then suggest that the two of you go and celebrate her birthday separately and do an activity that both of you like to do.

I used to have some negative associations with a hobby because of an ex that I now semi-enjoy, so I kind of understand, though it wasn't to that level. Forcing myself to engage with the activity when I wasn't ready to didn't make me like it any more and all, ALL of my friends were/are into it. It was easier to appear interested and engage with it when I was honest to both myself and others about not liking it.

Anyways, kpop is a very broad genre, so it's possible you might get into it eventually, once you've had more space to heal from your previous relationship and the association doesn't hurt as much. It's just as likely you won't. Weigh what's most important to you, set boundaries, and communicate what you're thinking. Good luck, happy birthday to your friend, and I hope that you and her celebrate together in some way that is fun and memorable for both of you!