Hmm...back in the psychiatric hospital? Well, she shouldn't be able to get pregnant there. (Yeah, I know, not the best solution...but it is something of a silver lining.)
The 5 year bc is Mirena. That's what difficult child 1 is looking into after she has the baby. She definitely doesn't want to have another one until after she finishes college. With Mirena, it is pretty much an IUD. You have it put in, then checked each year at your usual annual exam. It can be placed or removed at a routine office visit. So, if you did get her to agree, she could easily go to Planned Parenthood or some "free" clinic and have it removed without you knowing.
I think the only semi-perm bc (where she would not have the easy option of having it removed, refusing shots, etc.) are the implants. I don't even know if those are still available. (It's been a long time since I had to worry about bc...the implants were still "new" on the market when I had my tubes tied after difficult child 3.)
Would a call to your OB/GYN work? With the medications she's on, there would be issues with a pregnancy. Maybe he/she could explain to her what could happen to the baby as a result of the medications she's on/would still have in her system? (He/she would have to be careful, lest your difficult child decide to go off her medications for just that reason, but depending on how long her medications stay in her system, there might still be lingering risks/complications & he/she could explain the risks involved.)
I wish I could offer more help. Going through this with my own difficult child, I can understand your fears. My difficult child 1 wasn't deliberately trying to get pregnant - she was just plain old stupid & didn't use protection of any kind. But how you get here doesn't matter as much as just simply dealing with being here, if that makes sense. Luckily, the whole thing has scared the beejeebers out of difficult child 1. She's now darn-near panicing that she's going to screw up as a Mom...but I think it's a healthy kind of fear. She didn't mean to get herself here, but now that she's here, she wants to make sure she does the best she can.
I wish you the best of luck & will cross my fingers for you and your difficult child. I can only imagine how scared and concerned you are right now.