If your dd is in y5 is the other girl 10 yet?
if she is, you could also report her to the police...
I know that she is in foster care and a troubled child - however, maybe a stern telling off from the police (rather than anything going onto her permanent record, not trying to be vindictive and spoil things going forward) would help her to realise that she can't bully and attack and attempt to strangle people. It will also show your dd that you are taking this very seriously.
Agree that the teacher's behaviour is much worse, would definitely be reporting that and making sure there was a paper trail to back it all up and have it on record.
I would ring the school now and ask them to tell you where on their website their complaints procedure is - think it is supposed to be on websites these days although not sure if that's a legal requirement. If not, ask them to email it to you, say you'll hang on the phone while they do it and so you can check that it arrives safely (even if you have to say that you have been having email problems recently - it will force them to do it while you wait, rather than put it to the bottom of their pile of things to do that never get done). If they don't send it while you wait or give you the link to it, then follow up with an email to remind them to send it to you as you said you were unable to do this while we were on the phone. And then again about an hour before hometime, and then an hour after everyone gets in - keep ringing, twice a day until they send it to you if they don't send it out immediately.
Then they will know you are being serious and this could have serious ramifications for them...
Not sure if reporting an incident like this on the OFSTED website is the sort of thing that could trigger an unannounced emergency ofsted inspection - might be worth keeping in mind... 
Finally - I'd send a separate simple email in regarding the incident, asking what measures the school is going to take to ensure that your dd is safeguarded from both her teacher and the bully while she is in school, as it is evident that she has not been. Maybe say that until the school is able to reassure you that your dd will be kept safe and away from both these people who have caused her harm, she will not be returning (if this is what you want to happen - I don't know if you want dd to stay at home again, either in the short or long term). That you will be sending in a separate letter regarding the complaints and incidents, but in the immediate here and now, you are concerned that the school are not able to safeguard your dd. And that you are particularly concerned as one of the people that you should be able to rely upon to keep your dd safe is one of the people that not only harmed her physically but mentally as well by not keeping her safe from the bully and punishing her when she had been attacked by the bully. So that any absence that happens as a result of their failure to provide a safe environment for your dd to be in school is due to them - this is likely to hit their absence figures which as we all know they take seriously...
Think there is somebody at the council who is there to keep an eye on schools and absence rates - they may well be a good ally in a time like this if you approach them proactively (and you might already know them if you previously had problems and then HE'd) and say that you feel you have to keep your dd out of school to keep her safe, school are being unhelpful/unco-operative/etc and refusing to safeguard your child, what can they do to help as you want your dd to go back but safely without fear or anxiety.
However, the key is to keep using the word safeguarding (and pointing out their failure in that duty to safeguard) - apparently it's a word that they are legally required to sit up and take notice of - whereas if you were to make the same complaint but just say that the school didn't keep your dd safe or something similar - it doesn't carry the same legal overheads and the school aren't required to look into it and report it in the same way. Bonkers I know as within general speech it means the same thing but worth exploiting if you realise.