Ask for a meeting with the headteacher - do not be fobbed off with any of their monkeys, deal with the organ grinder. If you and partner can go s much the better, Ask at the meeting for a copy of the school's anti-bullying policy (they cannot refuse to give you this) and the behaviour policy. Ask what actions the school has taken re. the threats made in a teacher's hearing - ask the headteacher if they would feel content with this had their child been threatened in this way. Make notes (hence two of you) all the way through the meeting. Tell the headteacher that you believe the school is failing in its duty of care towards your child, and ask if they are content with this. Ask for assurance that this care will be properl given and make a record of the head's response.
Go through the anti-bullying and behaviour policies at home and highlight the points that show where your daughter has been let down ie the policy has not been followed or is failing.
Next step: contact head of governors and outline case simply, explaining that you are concerned that school is not exercising duty of care to your child.
Meet with a no claims-no fee lawyer and talk about your case.
This school's cage needs rattling firmly, good luck and look after your daughter.
Worst case scenario, you need to educate your daughter at home and you will need compensating for the cost of private tuition in key subjects, plus for the psychological trauma of the bullying. Do not put up with this and insist all the way on your family's rights being respected.