Pamzam101, Premier Inns are good hotels for those on a budget, yet are consistently clean with very comfortable beds. Premier Inn County Hall is very close to Westminster. You just turn out of the hotel, walk a block, then cross the Westminster Bridge and you're there. Near there are the Churchill War Rooms, Buckingham Palace, and Big Ben. The London Eye is actually just a few steps away on the Thames. So for most of your sights, that would be convenient. I stayed there last year for 5 nights and found it to be a very good hotel that was reasonably priced.
There is another PI just up the street (Premier Inn Waterloo) so you might want to check prices for both. Other Premier Inns in the vicinity include the Premier Inn Victoria (near Victoria Station) and Premier Inn Leicester, near Trafalgar Square. You will also see a number of Premier Inn Hubs, which are cheaper but much smaller (some rooms do not have windows).
If you're looking for something near Tower Bridge, I would suggest the Premier Inn Blackfriars, which has gotten good reviews here. It's relatively close to the Tower of London, etc., but also close to Covent Garden and very near St. Paul's Cathedral.
All PIs provide breakfast at an additional (reasonable) cost. And keep in mind that PIs offer a variety of pricing options: Non-refundable (the cheapest), a hybrid price that is payable in advance but you can cancel up to 25 days ahead, and completely refundable. Generally the earlier you book, the cheaper the price.
