Now you've got somewhere to spend your nights, what are you doing in the daytime?
London is expensive. This much is true and there are a good number of popular high priced attractions to lighten your wallet. But you can also have a good visit without breaking the bank. Get a good guidebook (borrow one from the library, or download and print lots of pages). There are some walking tour books that are nice, that gives you the chance to explore some of the lesser known areas of London off the beaten track. Some sites have walking tour podcasts you can load on your mp3 player or iPod.
Alternatively, take a walking tour with London Walks. I can't praise them highly enough. You decide which tour you would like and show up at the underground station/exit where the guide will be waiting. Pay the guide, usually about 6 or 7 pounds per person, and off you go. Most walks are about an hour and a half to 2 hours at most. They won't include transportation, on the off chance it's needed, so have your travel card ready. They also do day long tours which cost a bit more and you need to make sure you have enough cash for entrance fees should that be needed (I.e. The tour that does Richmond and Hampton Court also cost me the price of the entrance to the castle and the riverboat we took up the Thames but still all worth it to have the guide with us.) You might also want a few quid to tip the guide at the end. All the guides are Blue Badge rated and they are all excellent at what they do.
Here's another budget tip: some of the best known museums and a few of the lesser known ones in London are free. The British Museum, Victoria and Albert, Natural History, Science Museum and the National Gallery and Portrait Gallery are the best known. Each is huge and will take you all day if you were to do it all in one go. I wouldn't recommend it as you soon find all the exhibits and paintings blur together. They're definitely worth visiting, but get a map and decide what you would like to see the most. You can always go back. The Tate Britian and Tate Modern art galleries are also free.
Other museums that are free, are smaller and easier on the feet include the Wallace collection, the Sir John Soane museum, and the Imperial War Museum. Walking around Lincoln's Inn where the legal community is and through the Temple area is also a great place to walk and get away from the noise and traffic.
One museum that is a favourite of mine is the Museum of London. It's free and is located at the Barbican in the City of London. It tells the story of London from Roman times forward and has some excellent exhibits including one on the Fire of London. You can also see the gilded Mayor's carraige and lots of artifacts from many eras. There's another branch of it in Canary Wharf Docklands as well. This branch tells the story of London with relation to trade and shipping and commerce and is in an old sugar warehouse.
Another really popular event with local Londoners that not a lot of tourists really know about takes place over a weekend in September. It's the London Open House weekend. There are hundreds of historical buildings or buildings of architectural significance in London that are not open to the public normally but many of them are this one weekend with free admission thrown in for good measure. You can purchase a guide or search online at their website and then plan your route. Some may need reservations to be made or have timed entry so check first. I have a number of friends in London that go every year and find it really interesting.
The Changing of the Guard at Buckingham Palace is free and is a daily event in the summer at 11:30 a.m. It's every alternate day in the off season. The nearby Horse Guards also has a changing of the guard and it too is free. A bit less crowded and there's horses!
Althought Westminster Abbey and St. Paul's Cathedral both charge admission, they are worth seeing. For a free cathedral with just as many beautiful windows and frescos and a bell tower with an elevator/lift is the Westminster Cathedral near Victoria station. (The lift to the tower does cost a couple of quid). Lots of smaller churches are free and there are many gems around the city to check out, with long histories and exquisite stained glass. Free London Listings is a good site to find out about free or very cheap things and events in London.
There are quite a few high priced attractions in London but some of them might actually be worth the cost if it's something you really are intersted in. One place I've been to twice and still want to go back to is the Tower of London. Each time I've been there, I've run out of foot power before I've seen it all. The Crown Jewels are pretty spectacular but the rest of the compound has so much more to see. You can also join a tour with one of the Beefeater guards for free. They put on a good show and are very entertaining. The London Eye observation wheel is pretty cool. Even if you're not great with heights, it's not really too scary.
Hampton Court is outside of London and rather pricey, too, but if you love Tudor history, it's most definitely the place to go. I'd love to go back again. The Globe Theatre museum is also fascinating and I enjoyed Kensington Palace as well. I never bothered with Madame Toussaud's as, personally, I find that whole idea rather tacky but lots of people enjoy standing in long lines to get in and taking their photo beside wax representations of well known people past and present.
There's a lot of push for the London Pass card that gives you free access to many of the paid sites along with discounts to others. That might be an option if you are the type of person to cram a lot of things in one day. It does let you bypass long ticket lines but for me, I would never manage to get to more than two places in a day and it wouldn't be worth the purchase price. I'd rather go early to get ahead of the worst of the lineups or find places that have evening openings and arrive around 5 o'clock.