Well what I can gather from this is that a direct confrontation is needed. It's not helping any to just yell at them from your apartment because it seems to aggravate them so they want to retaliate. What I would suggest is to just march right up there and tell them what you have told us. If they don't get it at first, go collect yourself and then go back the next day. Tell them that you are paying rent and you have certain rights and they are not being respected.

I feel like they maybe don't understand what the problem is because it never has been addressed directly. You are assuming they can read your mind about your frustrations, when they may just think that YOU are being the problem by being a "cranky neighbor". There seems to be a huge breakdown in communication here, with both parties just getting mad and stewing about it and nothing getting done. At this point I wouldn't even worry about being nice about it, just tell them matter-of-factly that it is unacceptable behavior and it is bothering you and they have to correct the issue.

I wouldn't bring up the fact that you think she has mental problems. It seems like it would offend them and cause a further breakdown in communication. Perhaps you could make a letter that states something like, "I have certain medical conditions that require peace and quiet. There is a lot of noise coming from your apartment. Please respect my privacy and my health by discontinuing the noise. Thank you." and slip it under their door every time the girl starts making a racket. You shouldn't just have to live with it if it is causing you such distress. A solution needs to be reached because everyone sounds miserable.

As for CPS? Not a good idea at all. You have no proof that they are abusing her in any way and it would cause undue hardship and grief on the family, who seem to already be in distress.