Hi,
Just really see ALL the estate agents, it took me almost one year to find a 3 bedroom flat with garden in Clapham when I wanted to move from North London in 2009. Some of them did not register me, others gave up with my budget after a few viewings, other tried to convince me to buy a 1 bedroom flat showing me some flats which actually were so badly priced (they were actually on the market for years and may still wait for a buyer today
.
It has been difficult, but at the end I found within my budget which was similar to yours, I would only add that you have to check on a daily basis if anything new is coming up as the best offers sell really fast,
Good luck