In an emo relationship between a girl name Sphinx and a girl who is now a boy called Jay, and Jay breaks up with Sphinx and with that she threatens suicide and gets sent to a mental hospital, what is Sphinx supposed to do then?

First of all Jay is still a girl ... there is no such thing as girl becoming a boy ... just because she changes the way she acts does not change her gender ... she was born a girl and will die a girl.

second of all Spinx needs to get over it and learn that it is not worth killing herself over some person that never cared about her in the first place

sorry if this is not the answer you were looking for but it is the exact truth.

Talk to her and make her understand that its life