My only contribution to this is to say that
most agents will sell a house for 1%.
Foxtons are a particularly slimey bunch.
If both parties use the same solicitor, it doesn't half speed things up.
Fixed price conveyancing is the way to go. Use a firm in a remote part of
the country and they're grateful for a £300 fee. In London they won't pick
up the phone for that sort of money.