Hi there,
When I was your daughters age I was self-harming as well. I'm really sorry you have to go through this as I know it must be difficult. It's hard to explain why or how my own self harming came about, I can't really pin it down except that I remember doing it for a very long time.
I don't think it was just sadness that caused me to self-harm, but I difficulty in expressing and feeling my emotions. I was depressed as well, which sometimes aggravated the situation. I do think that sitting down and having a conversation will probably be hard for both of you.
why don't you ask her if she'd like to talk about, tell her you don't want her to feel like she has to talk about it, but if she wants to talk about it you are there for her? Maybe if you notice she isn't feeling happy talk to her about what exactly she is feeling and why. I had a very hard time expressing my emotions and I tended to lash out against myself or others. I'm not saying she is feeling the same, but it might be a reason.
I personally found cutting like an addiction. So I suppose you could treat it the same way you would any other addiction and that might help give you some ideas on ways to do something.
I hope I've given you a slight idea of something to do or say to your daughter. I do think, if your daughter wants to, you could go to the GP and ask for advice there and see if you could get her in to see someone other than family that she could talk to about her feelings, in case she might not feel comfortable doing it with you.
Sending positive thoughts your way.