Do you want to use the car in London (congestion charge and all?)?
You should be able to get something for £1000/week.
\link{http://www.homelidays.com/EN-Holidays-Rental/130_Ads/AdSheet.asp?PK=66721&FROMSEARCH=TRUE\Here} is one in Ealing that looks ok. 718 Euros/week. Two bedrooms, two bathrooms. One very nice thing about the homelidays site is, it now has testimonials! But none available for that flat. And all the other London flats either have only one bedroom, or one has six bedrooms and no specified price.
VRBO has a lot more places, but prices tend to be over £1. \link{http://www.vrbo.com/56027\This} one looks good, v. central. I think it's on the South Bank somewhere. \link{http://www.vrbo.com/69051\This} is similar. More Westminster, but prob still south of the river.
\link{http://www.vrbo.com/80711\This} one is near Portobello. Looks quite nice, up near the top of your budget though.
OMG \link{http://www.vrbo.com/76724\this} place looks great, and is a great deal, but a bit pricey for you maybe? Also only really available in holidays, I think it's someone's home, normally. Very beautiful, though! Oh, looks like one double bedroom and two singles, though?
3 beds, Camden Lock is nice, not expensive: \link{http://www.vrbo.com/77780\here}.
Ok, there are tons on VRBO and the front page makes clear how many bedrooms and bathrooms each has ... I'm stopping now as this is getting silly.