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$1,000 is indeed a tight budget even with the recent slide in the pound to the US dollar. Depending on the exchange $1,000 will work out to around £700.
I'm sure a hostel could be had for around £35-45 per night. Hotels will be considerably more. The further out of the City the cheaper it will be (generally) but then there's the transport cost (see below).
Whilst the museums are free entry, seeing a show in the West End will start at around £50.
Buying a Oyster card is a must but there is the travel cost with either the Tube or bus.
If accommodation does not take up most the budget food will. I would factor in a bare minimum of £20 per day. One could visit local grocery stores (Sainsbury's, Tesco etc.) at the end of the day and buy discount labelled food.
All in all, 700 quid for 10 days in London nevermind a side trip is very tight and not realistic unless the trip is done on bare bones minimum. A lot will depend on the cost of accommodation. Realistically, I would plan on £1,000 or roughly $1,400 and that will also be a 'just depends' budget.
H