Ypu should look at what you want to do for 10 days. London has lots and lots of great free stuff - but if you are not a museum person etc then you may want to think about getting a megabus to another city for a few days. You can get journeys for a couple of pounds each way if you book in advance and generally every other city in the UK will be cheaper than London and give you somewhere else to go out in such as walking in the countryside!
From my personal perspective I would struggle to stay within your budget in central London but that is my style of travel.