I'm sorry your family is suffering with this, especially your mom.
It really sounds like your mom needs to get back into counseling so that she can deal with her trust issues and other issues (insecurity, low self-esteem, her fears, etc.) and find a path that she can be at peace with.
Also, I think that she might need some psychiatric medication if she is hurting badly enough to want to be hospitalized.
If you can get away to visit her, do it. If you can't sending cards (actual cards through the mail), just a note so she knows that someone out there is sending her good vibes, can help.
Also, encourage her to get out and do things--it won't solve the problem but it will help her to focus on others instead of her own worries/problems. Something like taking up a hobby or volunteering with a charity, where she'll get out of the house and meet other people, can help. You actually don't want to take away her caring for the house and your sister, if she is able/willing to do it (if she's overwhelmed, then yes, by all means, do help if you can--maybe you could hire a cleaning service to help her 1 or 2 a month or something?).
If it were my mom, I would go see her. But I'm not sure I would leave my husband/family for 2 or 3 months to stay with her. I would see about lining up support for her (other family & friends to come over & help or to visit or to get her involved in activities outside of the house).
Also, think about this more like a medical issue, because yes, it's emotional pain, but she sounds depressed and anxious, and those should be viewed the same as a medical problem--get in & see the doctors/professionals who can help. The best thing is to get her to a counselor so she can work out what is the best thing for her to do... and no one can do that for her, much as you want to.