Yes, you DO have a responsibility to tell the mother.
Think of it this way: if, God forbid, something happened to this girl... say she was cutting and she cut too deep and died... and you could have prevented this by alerting her mom so she could get some help, you will never be able to forgive yourself, I would wager. And neither would your daughter. If you don't do it for your daughter's friend, do it for your daughter. Everyone deserves peace of mind.
When it comes to a potential issue of life and death, screw privacy. Screw confidentiality. I've had many people "betray my trust", and at the time I am so mad I could barely think. But now I understand that those "traitors" were very, very good and caring people. There's no conspiracy against me. People want to help me. I know this now.
It may seem a little unbelievable, but it very well could be that this girl may actually want, on some level, your daughter or you to tell her mom. Even though she may deny it until she is blue in the face. I know. I have been there. I could get very indirect and/or contradictory.
YOU MUST TELL SOMEONE ABOUT THIS! And please update.