You really need to give some idea of your salary in London.

You can happily pay £2,500 per WEEK for a 2-bed apartment in Chelsea / Knightsbridge, through to £250 in somewhere a bit grotty but OK through to less in a real danger zone / long commute.

Why a 2 bedder? 2 bedders are significantly more expensive than a 1-bedder of the same size, in my experience, and you have wasted space (in what will already be a modest space) when you have no guests.