I TOTALLY UNDERSTAND WHERE YOU ARE COMING FROM!!!
Just had to get your attention OP! :)
I am your daughter (not literally!) I had a private education from 5-16, my secondary school was a highly academic selective girls school. I did extremely well in my GCSE's and have, to this day, what I consider to be an above average intellect. BUT, to my detriment I decided in my 16 year old wisdom that I wanted to do a BTEC in Beauty Therapy! My Mother allowed me to choose my own path and I went ahead spending two years at college. I left got a job in Beauty Therapy, hated it, took A levels at night school to get a Uni place, went to University but I always felt 'behind' as I'd basically wasted 4 years of my life 16-20. I have never fulfilled my potential and I wish, almost every day of my life, that my Mother had insisted I finish my A levels and go straight to University. It is my one and only regret - I'm 40 years old and it is literally my only regret.
My own children are at an independent Prep school and I have always made it clear that their educational path is Prep school, Senior then University. After that they can choose for themselves but they need to complete their education in order to make an educated choice.
In answer to your question, I don't know how you can 'make' her do A levels and a degree, that is entirely dependent on your relationship and how good your communications channels are. I know my girls will follow the path I have set out for them as they know I only want the best for them and will ultimately support whatever they choose once they are in a position to make those choices with clear and sound judgement.
Good luck, you are just being a good parent by wanting the best for your child.