AlethiaSi,
I'm really sorry to be so blunt, but it sounds like you all didn't think through what you were doing. When you set up "fuzzy wuzzy" arrangements, you can't be mad later when the other party oversteps the fuzzy boundaries.
If she wasn't a sister, why did you let her wear your letters? Doesn't that now give her the impression that she's one of you? I don't blame her for thinking she could do the things you describe (go to events, etc). How come nobody said, "Hey, I know you enjoy participating with us, but it's not appropriate for you to attend our meetings until you're accepted for membership"? Because you all didn't think to make the boundaries clear, you're now going to have a difficult time asking her to step back (or away).
I don't mean to be so harsh, but sometimes we need to take these things as learning lessons. You say she needed help and you gave her assistance. I know you're disappointed in her now, but you willingly gave her your help, right? You're now mad at her because she isn't following your plan? It may suck for you, but I don't see how this amounts to a "betrayal". You gave, she took.
As for her sleeping around, that's not your fault and your sisters shouldn't be giving you grief about it. So what if you invited her along? It sounds to me like they willingly played along with this arrangement.
Perhaps it's time for you and some of your officers to sit down with her and explain to her that her behavior has overstepped the lines of good taste and that you will not be considering her for membership. Then kindly ask her to not attend anymore of your events, ask her to move out of your house, and tell her to get her own ride to work. It's gonna blow up, but you all need to pay that price now if you want things to get fixed. Sorry, I wish I had better news for you.
.....Kelly