You can do it easily.
If you book well in advance, you can get some great deals with Travelodge. Some hotels are cheaper than others; Aldgate East is right next to the Tower of London but you can get rooms for under £50. City Road is good value, too. Kings Cross Royal Scot is dirt cheap, but avoid it - it's awful. If you're willing to stay a little further out, Bethnal Green is extremely good value - it's still only zone 2 though with easy links to the west end etc. When you book, google 'Travelodge voucher codes' or sign up to their mailing list beforehand - there are often really good discounts available. I think at the moment there's 20% off with SUMMER2016.
The easiest way to save is on transport. While you're there, use an Oyster card but only travel on buses - your fares will be capped at (IIRC) £4.60 a day, however many journeys you make through however many zones. Buses are much nicer imo - you see so much more. The 11, 14 and 15 routes are pretty much the same as the expensive bus tours - they go past the Tower, St Paul's, Parliament, Whitehall, etc. Plan like a demon beforehand, using an app like London Bus Checker and Google maps to see which buses you need and which routes from your hotel etc. I like that part 
While you're there there's so much free/cheap stuff to do. The British Museum, National Gallery, Imperial War Museum, National Portrait Gallery etc are all free. Also lots of quirkier little places like the Sir John Soane's Museum in Lincoln's Inn. The Guildhall with its Roman Amphitheatre in the basement and the Museum of London are next to each other (near City Rd Travelodge actually) and are unmissable. MoL Docklands is fantastic too. Also Greenwich is definitely worth a visit - the Old Royal Naval College is stunning and was the set for lots of films. One thing you should definitely do is try to catch a performance at the Globe - 'groundling' tickets are only a fiver and it's the most fun, immersive theatre I've ever experienced.
What I'd say is do lots of research - what's your 'thing'? History, art, music, landmarks? There's so much to do for little or no cost if you know what you're looking for. Definitely avoid expensive tourist traps - Madam Tussaud's, M&Ms world, that kind of thing. Just a walk along the river on a beautiful evening is lovely. We grab food at a Wetherspoons or similar, cheap and cheerful. There's loads in central London.
I think I'm going to have to book another trip now! 