Check out Gumtree and Rightmove. That should give you a fair idea. Knightsbridge is really not the place you want to be looking on your budget. Hammersmith is much cheaper but still relatively expensive given your budget. It would be like looking for a place on Park Avenue in New York on $75k/year. I'd consider North-West; Queens Park/Maida Vale. Or something south of the River (Clapham/Wimbledon/Southfields), or East London. Otherwise I'd suggest going further out of London, where most things will be cheaper not just rent. For two people on £50k/year I'd be looking at something around £250 max/week, and even then you won't have much (if any) money left at the end of the week.

I'm not trying to discourage you, I'd highly recommend the experience.