OP I am aghast... Really I am...
What I am aghast at is not the school but you OP
Schools get it wrong / don't consider what pupils might need and actually they rely on parents and pupils to discuss issues with them so they change things... If they refuse to work like that then that is a whole different matter.
Anyway my biggest issue is this OP - what you are doing is reinforcing the feelings of shame & embarrassment that your daughter feels about her period.
I was a midwife and worked in gynae for many years - I see the other side of this 'embarrassment' being reinforced - woman completely ashamed because they are bleeding or need examination.., No woman likes it but I am talking complete & utter shame that their female body is working in the way that it should.
Keeping her off school solved the issue that day and is fine but doing that and not finding a solution is you saying to your daughter 'Oh my god you are right we MUST keep your normal bodily function a complete secret as it has the power to ruin your world'
I get that you are both a bit embarrassed but someone suggested the school nurse getting involved and this is a good idea but will take a long time given that school is about to finish for the summer.
Whilst your daughter doesn't want you to talk to school the present situation is one that has potential to just continually reinforce the feeling of shame she has about her period. If she leaked other children probably would laugh and again that feeling is reinforced. What about residential school trips or activity days? Without solutions / coping mechanisms she will just learn that having a period stops you doing things.
Go to your daughters school and ask the questions - you can simply say you thinking ahead and your daughter hasn't started her periods yet but you wanted to prepare her.
And if school refuse to help - I would write to the Governors if I was you but then I have strong feelings about women loving and understanding their bodies 