Why are you writing a letter? It would be much easier to phone and ask to speak to them or to pop-in and try to speak in person.
For work-experience in a GP, you tend not to require anything apart from a CRB (some GPs are OK without it) and sometimes an informal interview.
As for what to do, you will most likely end up doing admin or secretarial work. If you're lucky you may get to learn the basics of clerking. I only sat in with the GP a few times during a week's shadowing and most patients kicked me out regardless; as such, without any medical knowledge, this isn't useful anyway. So in essence, you want to learn the importance of primary care, how it works, what doesn't work, what needs changing etc... There is a lot you can take home.