Have not yet read the other answers, will do so after posting.
Because this friend is mentally ill, you have to rack up some/most of her meanness to that. Actually all of it, really. Because she can’t (most likely) have a rational conversation at this point.
But, that does not discount the fact that she said things that hurt your feelings and upset you. I am dealing with something somewhat similar right now, and until you wrote this question, I have been feeling guilty about feeling terrible about how I think I was treated. I got an apology that didn’t apologize for what was actually said and done. It was sort of like, “Sorry, let’s be pals, I missed you, blah, blah, blah, how’s the weather?” She never apologized for what she actually put me through. But, I realized that she went through some very serious trauma (nothing to do with me whatsoever, but she moved away and I didn’t hear one word from her for almost a year).
If I were you, in your particular situation, I would very gently blow her off, repeatedly. You don’t need to be mean, or even explain to her why or how she hurt your feelings. Either she won’t understand, won’t remember, or won’t even fathom why you are upset. She gave an apology that doesn’t really fit the circumstance, which is kind of sweet, and kind of messed up at the same time. You have acknowledged that she is mentally ill. You pretty much have to go with that. She can’t really apologize for what she did to hurt you, because she probably doesn’t perceive or even remember.
She does seem to care for you and like you, which is good, but that doesn’t mean that you have to give in to her or coddle her. It might be different if she was a family member, but she isn’t. All you need to do is be a decent person and maybe not chew her out (even if you want to do that, and even if she deserved it as a sane person). You need to be kind, but distant.
Let her know, again and again if you have to, that you don’t have time to get together with her. You don’t need to give her any explanations. Explanations will just make the whole situation worse. Move on from her, slowly, but surely.
I feel so sorry for you, because I know how it feels, even though our situations are quite different. It would be so easy just to tell her off, but that would not serve any useful purpose. Be gentle, but move away from her. And don’t feel guilty.