I would get help. Now. Ask the school to support another CAHMS referral, make it clear that dd is a definite physical danger to herself and others.
It is very bad for for her for this to be allowed to continue unchecked and sooner or later she will get too strong for you to restrain. She needs help before that happens.
Dd was never an unpleasant person, but she did go through several years of violent tantrums which sound very similar to the ones you describe. In her case, it was to do with past trauma and ongoing physical and emotional pain. We have spoken of it later and she has confirmed that she would go into a state where she would quite simply not recognise me as her mother any more; all she saw was a dangerous monster that she had to fight off.
One thing I did do that I think was helpful was to make sure she was never unchecked when trying to hurt other people. It is very, very frightening for a child to feel that there is nobody there to protect them from the rage that is inside them.
After the first lashing out, I would get behind her and restrain both her arms using the leverage to pull her teeth away from my hands. I would back to a chair and sit down, putting one leg around her legs to keep her from kicking and keeping the other one on the ground. All the time I would repeat as calmly as possible "no, I won't let you hurt anyone, no I can't let you hurt anyone".
I sometimes wondered what I would do if the tantrums didn't stop. And I was quite clear in my mind that if that happened I would have no alternative but to ring the police. The one thing I couldn't let happen was to let dd continue unchecked until the day she killed me or put me in hospital: that would ruin her life as well as mine.
But long before that happened, dd was under the care of CAHMS and the tantrums had stopped.
My db went through a very similar pattern, though there adoption trauma was probably the initial trigger. He is the least violent adult you could imagine.
As to discipline, I and my parents took a very similar approach: we kind of accepted that on a bad day meltdowns would happen whatever we did, and kept on enforcing discipline the rest of the time.
As another poster said, this behaviour is often anxiety related. Your dd needs to see that she can get help to get her anxiety under control. But also that she cannot manipulate you, that you care enough about her to make sure she gets help whatever threats she makes against you. You need to pretend to be totally unphased by any threats to hurt you again, to run away, to lie to others about you. You need to tell her that she is seeing the doctor and that is that.
But also make it clear to her that they will be there to help her to take charge of her own life, so that she gets more fun out of it and has more control.