It sounds like your sister has taken this out of your hands. She probably has some deep-seeded issues about the loss of your mother that she's never properly dealt with. If she's not in a mental place to have a relationship with you, the best you can do is let it go. She's your sister and will always be your sister. If she gets her act together, she may come to you in time. That doesn't mean you have to resume the role of her caretaker (that won't help your relationship as sisters.) If she is cutting off contact with most of your family, it is definitely not just an issue she has with you. Sounds like self-destructive behavior and that she is not in a place to accept help from her family right now.
Also, I am not clear by your post. Did your sister have a sexual relationship with your boyfriends during your relationship, or after you had broken up. It may not seem like much of a difference, but it does speak to whether she is behaving out of spite toward you or if she truly believes she deserves no better than your hand-me-downs. Something to think on.
Either way, you can't expend excess energy on this. She must be allowed to live her life and make her mistakes. I just hope you can take her actions and inactions in context and realize they are born of some sort of pain..... and perhaps take a moment to feel sorry for her, then move on to making your life the best it can be.
Congratualtions on ditching the jerks who seem so willing to use your sister for sex as well as emotional manipultion of you.