At OP,
to London or not to London is a long-running debate. It can be summarised by the question 'is London worth the extra expense?'. In general, you only get 1000-1500 extra student loan for living in London, whilst your accommodation cost will be, at minimum, 2000 more expensive than most students (not talking about halls, but for your other four years). London in general costs 2-3k, average 2.5k more per year than at any other university. You'll pay 150 a week, maybe a bit less if you're lucky, for accommodation, perhaps a bit more for halls. Outside of London, in halls you'll probably pay 100-150, and the rest of your time probably 80-100 in Birmingham.
The truth is that it's difficult to say whether the expense is worth it. It comes down to your personality. If you're outgoing, love trying new things all the time, going to new places and love the added element of prestige that often comes with activities in London, then you will love Kings. If you don't know right now, then I'll bet that you'll probably love that X factor that stuff in London has. Everyone I've known does. If you're a bit simpler and don't feel you need that extra buzz or whatever you want to call it, then maybe Birmingham would be a smarter choice. London is truly magnificent though, and there's no way anybody can doubt that really.
Birmingham is a good university with an A grade medical school, I've got quite a few friends in 4th year there and they've had a good time, there's a great vibe and a good environment. I struggle to believe that their teaching experience is better than Kings' though. I know about 30 people at GKT, from all different backgrounds, in different friends groups and societies, and they've had the time of their lives. There's a reputation in London for GKT students having more fun than UCL, Imperial or Barts, and I'd guess that. Kings has a magnificent student environment and having studied at a different med school in London, I'd back that up.
Overall, I'd say Kings shades Birmingham. The uni environment is slightly better imo and London blows Birmingham out of the water. There is extra expense involved and I would suggest that London costs 2-2.5k more than Birmingham in a given year. It's up to you whether it's worth it.