Sending her to juvie/jail is risky because it could push her completely over the edge, make her hate her parents, and connect her with more psychotics.
It all depends on her reasoning. You have to be good at reading people to know if they're being overly dramatic, if they're slightly delusional and cocky, or if there really is a screw loose.
If they're being overly dramatic, as high school girls tend to be, airing out grievances should help.
If she's delusional and arrogant, you need to shock her system aka "scare her straight". Maybe make her realize she never wants to spend time in juvie/county jail again. Teach her respect, aka never talk about her family members that way again.
If she's really a little off, it's tougher because the entire mental health game has changed. They don't have asylums anymore, they just drug people up. I don't think she is, though. She's either on drugs/pharmaceuticals, was molested/assaulted early in life, or has some extremely skewed/warped view of her current family life.
It all depends on a frank confrontation/conversation and seeing if she's the type of person who can be reasoned with, or if she's a sociopath and will lie to you no matter what.