The school isn't going to want to acknowledge that it does have a bullying problem... because, if they do, then not only will it make it/the staff look bad, but it could also potentially prevent other parents from sending their children there in the future. I think the same can be said for every school, to be honest, and I understand why. Parents have power, at the end of the day. We communicate about schools with other parents and, as such, can influence others decisions as to whether they want their children to go there - or not.
However.
Your daughter is being assaulted by another child. Let's be frank about this. You said: 'We've also had her pushing our dd off a chair during a lesson, shoving pencils in her ears, pushing her to the floor then dragging her around a corridor etc etc.'
That's assault.
Pushing her to the floor, then dragging her around, pushing her off a chair... that's physical assault, and will have been very frightening to your DD... but the bit which really frightens me is that this child has pushed pencils into your daughter's ears! The potential consequences of this are horrific!
If a woman posted here saying that her partner or husband had done this sort of thing to them, it would be classed as domestic violence! There is actual violence being used against your child - although obviously not domestic - and the school aren't going to want to address it for the reason I stated above. So.
You make them address it.
You need to go over their heads. Report this to the LEA. Report the school to Ofsted. If necessary, and the school still refuses to take this safely and protect your child and others from this child (who isn't angelic in the slightest, and I understand that parents can often be blind to their children's faults... but really? Where is she learning this behaviour from?!)... because your DD won't be the only one, and this behaviour will escalate the longer it is left untackled... you report it to the police. Who will - I promise you - make the school take it seriously. You have to protect your daughter. Show her that you will do everything in your power to keep her safe, and you'll teach her that it's okay to be lovely but that it's also necessary to stand up to those who hurt us. You'll show her that she's loved, and valued (which I have no doubt that she is... but being bullied can often cause us to feel that we're not worth protecting, if we confide in someone and they do nothing to protect or help us). That alone will give her confidence in herself.
My DS had a bullying incident at his "we don't have a problem with bullying here!" school. When the (useless) headteacher refused to do anything, we reported it to the LEA (who were brilliant and very instrumental in encouraging the useless head to retire; the new headteacher is the best thing that's ever happened to our little village's school) and when he was told by the Y6 boy that he was going to kill him, the police were also fantastic. They made the school take it seriously, and the little boy who had physically assaulted my son was given the actual help which he needed. Sadly, he didn't have the happy homelife that his parents made out... there was a lot going on behind the scenes, which made him "act out" for the attention it got him. But if it had been left, he may well have ended up seriously hurting someone - and I was adamant that someone wasn't going to be either my son, or on my conscious!
In essence, by not forcing the school to actually perform their full duty of care towards your daughter, you're storing up trouble for both girls, I'm afraid. You don't know why this girl is hurting your daughter... you may not actually care (perfectly normal not to)... but you can stop it from continuing. And your daughter will appreciate it. Maybe, one day, the other girl will, too.