You might benifit from more opinions by posting this on the mental health board. I think it's a very good sign that's she's told you and has asked for help. Given her age, is it GCSEs stress? Is it superficial cutting? If so some practical thoughts...
Accept what she says when you talk about it, be understanding, empathetic. Don't talk about it's impact on you. Keep everything focused on her. Don't blame or make her feel guilty or that she's misbehaving. Tell her you love her. Tell her you would like her to stop, accept her saying that she can't.
Talk to her about keeping herself healthy - making sure whatever she's using is clean, that she has plasters, antiseptic wipes etc.
Talk to her about alternatives, common advice is snapping an elastic band against your wrist if feeling pain is helpful. Or drawing (or temporary tattoo) something pretty where she cuts (eg a butterfly) when she feels well, so that she has an interuption in the thought process when she wants to cut when she's feeling low. Or talk to her about recognising the low feeling she has and doing something different with it eg talking to you, going outside, phoning a friend, headbanging to heavy metal. Whatever she would find to distract and help take her out of her feeling.
Restrict her access to anything that she can make a serious attempt of self harm/suicide with. Eg put the sharp kitchen knife and paracetamol in your locked car boot
A psychiatrist is not necessarily the right professional to see (I understand that's what's available on your insurance). They could possibly medicate her if she's anxious or depressed (but she's a child so might be very reluctant to perscribe antidepressants). But she said she feels low, medication can blunt the feeling, but not resolve the cause. Therapy does that. If she's wanting therapeutic help, I would recommend an ACP registered psychoanalytic psychotherapist. These are the only MH professions to be trained to a doctoral level to solely work with children. They work within the NHS/privately. If you wanted to investigate this option privately the website below has a search tool to find a local therapist.
https://childpsychotherapy.org.uk/home-1
You might want to start the ball rolling with a CAMHS referral via your GP
If she seriously hurts herself or says she wants to, take her to a&e