looking for a place to stay (London) 01.09-22.09

have a look at airbnb for cheap accommodation. You don't need two weeks to do all the above.

You could also look at couchsurfing.com and ditto, probably a week is enough, unless of course, you want to do some sight seeing too.....

Contact :http://www.posk.org/

Budget Guest House is an affordable accommodation located in the superb King’s Cross area, next to some of London's most eye-catching attractions. This budget hotel is a destination for all tourists who come to London to visit its fabulous landmarks. Victoria and Albert Museum, Madame Tussauds, British Library, British Museum, Tate Modern, Science Museum and Natural History Museum, are all within easy reach from Budget Guest House.
Plan your holiday in London at least 15 days ahead and take the advantage of our fabulous promotion: Book 15 days prior to your arrival and get 5% discount.
Tube Station: 7 minutes from King's Cross St Pancras 23, Swinton Street London WC1X9NL

1 mn from South Ealing Tube (Picadilly line) in Polish Area : polish shop in front with adds for room rent ...
7mn Bus 65 or (15mn by foot) to Ealing Broadway (Central Line) and Railway station to Heathrow and Paddington.

http://www.bennysbandb.co.uk/

www.couchsurfing.org

Are you sure you can open a bank account before registering? In my experience registration is done at the beginning of the term, although it might be possible that your university will allow you to register earlier. In any case, after registering you will get a letter from the university confirming your student status which you need to open a student account. For all other accounts you need proof of accommodation which I assume you do not have.