I only read your brief summary. Fyi.

This is not a friendship; it's a power play. Threatening suicide is a form of emotional abuse. Rather than a sincere request for help, it is a tool to manipulate and control another person. This is unacceptable in any situation and not your fault. I assume this person is an adult. If she is, then you need to just cut ties with her. If she is a minor (under 18) you can call the police and say you think she may be a danger to herself. They would likely bring her to a hospital.

Adults are responsible for their own mental health.

Please don't allow yourself to be abused by this individual anymore. Peace and hope to you.