It's good, but IMO you don't need to mention where you're going to apply - it has no real relevance, especially since you haven't even started applying yet, never mind received offers, so it could look like you're trying to show off. If it was me, I would write the letter like this (but everyone prefers different things so if you think mines crap thats fine lol)
Dear xxxx,
My name is Ninja32 and I am currently a year 12 student at XXX school, studying for AS levels in Maths, English Literature, Business and Economics, Psychology and General Studies. I hope to achieve 'A' grades in all of these subjects, and plan to continue them all/xxx to A2 level. After completion of my A levels, I hope to study a law degree at university, with the intention of becoming a solicitor/barrister.
Before applying to university later this year, I feel it would be invaluable for me to gain experience and further knowledge of the legal world by carrying out a two week work placement at some point between x-x, and was wondering if it would be possible to carry out such a placement at your firm.
I would be extremely grateful if you could get back to me as soon as possible,
Yours,
xxxxx
I would also maybe consider sticking an extra paragraph in each individual letter about the particular firm it is for, saying why you want to go there. For example, if it is a firm which specialises particularly in criminal law, say you have a great interest in criminal law or whatever. I would also advise you concentrate more on applying to smaller firms, because larger ones will probably have work experience opportunities specifically for uni students, and will be unable to accommodate you as well.
Hope some of that helped!!