I'm assuming this is your step daughter we're talking about, as per your biography.  I think it's totally right that you and her mother are having rules etc. - whether or not she has attempted suicide it will not help her to be allowed to use this as a threat every time something happens that she doesn't like. If you feel committed to this relationship, it might help to have some family counselling which the whole family attends.  She may well have mental health problems but she is using them to grab power over you all which is unacceptable. 

These kids are tough to deal with. Make sure you have a good therapist who works with both the parents to help you understand when her behavior represents real risk and when it's manipulative (hint: I'll kill myself if you don't let me do x,y, z is manipulative) and how to deal with it in both scenarios.            

I'm in the child mental health field and have been for 30 years, and kids are sicker now than they've ever been. Your daughter/stepdaughter's presentation is a common one nowadays - suicidal, self-harming when they are very young.  It can be managed with the right treatment team and the right parental actions.

I blame social media and endless internet access, personally. If she's glued to social media - cut her off. Yes, she'll scream about it, and use your crisis resources if she threatens suicide. But social media is toxic for girls like this.

She should be seeing a therapist who is teaching self coping mechanisms as part of a holistic process. If she is not being taught any self coping mechanisms, at some point you may wish to change therapists. 

That will help prevent a major relapse in later life once she has recovered from this episode.