H.W.
I think you need to just keep an open door with your daughter. Were I in your place, I'd try to be patient. Kids need help figuring friendships out, but they need to do it in their own time. A girlfriend and I were just talking about this in regard to our own kids. How do we help our kids see when they are being mistreated or their good, sensitive or caring natures are taken advantage of?
She and both remembered how we had to come to our own conclusions, to struggle internally a bit with what we felt we 'owed' our friend vs. what we felt we owed ourselves. For some kids, this takes a long time.
Last year, a wise woman said that, when it came to our kids and their interactions with other kids, we should try positive phrasing. One example: if a child was afraid of dogs, instead of giving that attention ("Oh, I know, you don't like dogs" or "Why are you so afraid?") we can say "I'm looking forward to the time when you are comfortable around dogs."
"Sweetie, I'm looking forward to the time when you feel comfortable letting Kay take care of her own feelings."
"I'm looking forward to the time when you can play with your friends and enjoy yourself without worrying about Kay."
More positive statements don't put Kay down, and they let your daughter know-- in a few words-- that she has permission to emotionally let herself off the hook with Kay. I would continue to allow your daughter to figure this out on her own, but it's also okay to give her some guidance too. Noticing that Kay falls apart in group settings, you can notice this aloud to your daughter the next time she considers inviting Kay to something; "Kay seems to be more of a one-on-one kind of kid." Encourage her to spend time with friends she enjoys more and let her know that it's okay if friendships fade. People change and she's going to have a lot of friends in her lifetime. And let her know you are looking forward to the time when she doesn't feel so worried about making her friend happy. At some point, it's up to Kay to make Kay happy. :)