It is hard to convey everything here about our loved ones properly in a few words but though I may be wrong about this because I may just be projecting, my take is:
I am sorry she was so insensitive to your feelings--she should have acknowledged your hurt feelings but also--isn't K her daughter? Sometimes people get really defensive about their children for complex reasons (I know I do) and it is true that neither she nor you are likely to change K's behavior. I am sorry she was so badly behaved toward you. I am sorry your sister got so angry at you for her daughter's bad behavior. I have done this to my husband; in the sense that I have blamed him too much for my son's bad behavior, not wanting to see his behavior for what it is.
Your sister may not be her daughter's issues objectively. This will not change over night. Until she sees it for herself (you can't make her because of all the emotions going on between the two of you right now), perhaps you two just need some space for a while. It might be good for both of you. I bet your relationship will turn around some day. You may both have some emotional issues.

You have said many times that your mom will never change also. I am sorry your sister's got so mad at you when you just needed to vent and get some compassion about how she treats you.

I am sorry she has been so cold about it. Is she always? Do you think that they are just tired of you ruminating about it over and over or do they never take your side? Sorry if I am not being that careful about your feelings. You have said you are still hurting over this.