I'm sorry OP, that sounds like you have a huge amount on your plate to deal with, and little (in fact no) support.
I know you said you're a private person irl but maybe now would be a good moment to start reaching out to your friends? I would hate to think a friend of mine was going through this and keeping it all to themselves.
Regarding your DD, please please contact school and ask them for help. They will be able to refer to CAMHS and any other appropriate support services.
I know it differs by area but I can say when we lived in East Midlands, CAMHS were fantastic and honestly if not for them I don't know if I would still have my son today. He is now a big burly bearded 20-something :)
It's hard not to react badly to self harm. Your gut instinct is: See wound, make wound better, kill human who did it, raaargh. Except when it's the person who did it to themselves, it throws a spanner in the works! Have you spoken to your DD about the times before when you didn't react the best way? Have you asked her directly "when you feel the urge to self harm, what can I do to help you?" For example, going out for a walk or a drive, going shopping (for boring shit, not wasting money on needless crap!), playing a sport you both like (even just playing catch with a tennis ball), cuddling on the settee and watching a favourite film.
You asking her this will encourage her to think about ways she can distract herself from self injury. It will make her more emotionally literate going forward.
I found my son is best helped by him helping others. So for example if he's having a low day, a great way to help him is saying "Hmmm, my wardrobe/kitchen cupboard is full to bursting! Let's organise a charity shop donation"
We also say sometimes "Hey it's been a good month, let's look up what the local food banks need and stick it in the collection box"
Last time we did this, a lady from the local food bank happened to be there at the time we were dropping off, and her heartfelt "thank you" meant so much.
Sorry I have waffled on but I hope some of this might help you with your DD.