Emil, haven't a clue for short stays.
Gumtree would be a good bet, but be aware, most adverts are from estate agents, not private.
When you coming? You should try and coincide your visit with a Free event like Notting Hill Carnival or Rise Festival.
Free Summer Festivals in London – London Summer Festivals, Free Festivals in London
p.s. We pay £850 pcm for a 1 bed flat, but it is a nice area with good links and amenities.