I self harmed as a young adult (19 to mid 20s,) it was worst aged 19-22 until I got on anti depressants
I still don't know if my parents know, I've never spoken about it with them. They both knew something was up I'm sure but not what I was doing.
Self harm is a coping mechanism. If she's been doing it s while it may well be her only coping mechanism now. The best thing a counsellor could do is help her discover other coping strategies that work for her. This is imo far better than talking through what is causing each cut kind of thing.
As a teenager I was very articulate, in writing anyway, about my feelings. I still feel that self harm took that away from me and I still struggle to express my feelings or even define them to myself. Any professional asking me what's wrong isn't going to get much of an answer.
I think your dd does need help from a professional, whether through counselling, medication or both. But there's very little point unless she wants that help. Self harm is a crutch and its a scary thought to have to give it up. I don't know how you'd feel about it, but could you let her know that speaking to someone doesn't mean you expect her never to cut again just because she's seeing someone?
I'm sure you're very upset about what your dd is going through, but try not to let her see anything other than that you're concerned for her. If she's worried she's upsetting you that's just one more thing to feel bad a out iyswim ?
Do other stuff with her, don't make every conversation about her self harm and other problems. It's all very overwhelming and can feel like its taking over your life. So give her some normality back. Play a board game as a family or go to dinner or something.
Possibly online support groups may be helpful, but proceed with caution as it may be too distressing and "triggering" for her. But it is good to talk to people who are going through the same thing, maybe easier than talking to a professional.
Sorry for rambling. I don't really know for sure what I would've wanted from my parents do just a few things that may or may not help I guess