My DS1 started using a laptop in school after his dyspraxia diagnosis at a similar age. He has now been using it in school for over four years and took most (but not all) of his GCSEs on computer, which is one of the main advantages of your DD starting to type now.
The way we did it was:
He went to a touch typing course in London for dyspraxics and dyslexics, but I believe there is touch typing software available
We set up standard documents with formatting in place so that he could start notes in a lesson or homeworks quickly
He started using it in lessons gradually: English and History first, where there is the most benefit. Before using it in a lesson for the first time, he asked the teacher if that was OK (of course they said Yes). I think he has to remember to sit near a power socket in classrooms.
He also started typing all the homeworks he could at the same time, For him, that was very soon much quicker than typing, plus it freed up some 'thinking space' in his brain.
Over time, he started typing in other subjects at school, e,g, languages, Geography - any essay-based subjects. He doesn't type class notes or homeworks for Maths, Chemistry or Physics because it is too tricky and they are not very wordy subjects.
I agree with a1b2 that the Dysraxia Foundation website is a good resource. Your DD might also want to go on the Dyspraxic Teens website, which is a forum which does what it says on the tin.
And does your DD have a SENCO at her school? She might well get some help from the SENCO with getting the laptop use sorted out.
The only thing is can't advise on is the self-consciousness thing. My DS1's version of dyspraxia means that he is not very aware of what others think or body language, so he has always been really happy to use his laptop. And it really made a massive difference to the quantity, quality and legibility of his work.
I hope that helps. Do ask me if there any other questions that you think of.