There are many reasons why people feel guilty for things, and often it's because they think they are either responsible for causing it or responsible for fixing it. If your daughter is mentally ill, you can and should be sympathetic, but you didn't cause it and you cannot fix it.
If your daughter was really thinking about checking herself into a facility, I'm not sure why you decided to talk her out of it. Did you think that putting her in your mother's home with a sister who is 20 years younger would fix it? I think you might reconsider that - consider supporting her as she seeks more intensive treatment. She's got quite a burden to carry, but it sounds like you wanted it to be a family fun fantasy. That's unrealistic. You not only equated a visit with you with intensive, focused therapy when she was so down, you kind of encouraged her to run away from her problem and then you put her right in the situation that had been tense and problematic before (life with you). I'm not saying you caused the initial problem with lying and stealing, but you invited her into an environment with pretty bad memories for both of you.
Then you turned her illness into a "vendetta" against you. That shows a fundamental lack of understanding of mental illness (and you say she has more than one). No one enjoys being in the company of someone who is verbally abusive, but to turn it around as an attack on you is a big stretch.
You invited her, so you should pay for the travel - not expect her to be overly grateful for it. One would think that having everyone together was your idea of a gift to you - perhaps in terms of help with your mother, perhaps in pretending the past problems were over with, perhaps something else.
I think you should not try this again, and perhaps should ask your older daughter if it would help for you to attend a therapy session in her town so you could better understand her issues. You could get some respite care for your mother or perhaps have your 24 year old daughter be in charge of Mom for a few days. But mostly I think you should work with your own therapist to work through this guilt as well as the simmering resentment from a daughter who lied and stole from you.
Mental illness (especially in combination) are not well understood in many cases, and it can be difficult to find the right combination of talk therapy, coping skills and medications to find relief. You don't have to be with someone who abuses you but you really should achieve a better understanding than what you have, and develop a strategy for keeping your distance that you work out over a period of time with a trained professional.