We spent 3 nights in London back in September and we stayed at the Tune Hotel Liverpool Street
http://www.tripadvisor.ca/Hotel_Revi...n_England.html
The hotel was fairly cheap ($120/night) and we booked through Expedia. The reason why the Tune Hotel chains are cheaper is because they cut out a lot of the things that most hotels would include (ie cleaning of the rooms only every 3 days; towels, wifi, safe, tv, toiletries are all at an extra cost). Since we booked off of expedia, all of the above extras were included for free.
Location wise, it is super convenient. The hotel is located in the East End of London (near the Gherkin Building). The Liverpool Tube station is only a 10 minute walk away. Brick Lane is a 10 minute walk away as well, and Spitalfields market (it is actually open 7 days a week) is only a 5 minute walk away. There are tons of restaurants nearby and public transit is really good as well, so you can get around pretty easily.
We watched Les Miserables and it was absolutely amazing. I would suggest you goto the half price ticketbooth near chinatown to buy some discounted tickets for the same day, then head into chinatown for some dimsum or bbq duck (which apparently is world famous).
Have fun! London is one of my favorite cities in the world.