I agree. We stayed at the Premier Inn near the London Eye. Great location on the river and across from Waterloo Station. Within walking distance to Parliament and many other attractions. The breakfast was good.
When we’re trying to see as much as we can the hotel is secondary to the sightseeing.

Dirt cheap
 

Location counts tremendously in London. You should know how you're going from the airport and try to get something nearer to the tube or attractions, many hotels that are not chains have very small rooms to European standards . You should check out the room sizes on several of the websites as well as making sure they have elevators and a breakfast room. Also we like to have availability of a supermarket / food store nearby if we want to eat in our room and some people do not appreciate English food.

London Hotel
 

I just got back from a month in England and have about $400 in British Pounds left. Would you like to buy them without any exchange fees?

Do not remember the hotel name but that is where we stayed last time. European hotels so much more charming than American chains.

Bankside
 

I personally like Bankside. Very central. Walk left along the river and get to Big Ben. Walk right along the river and get to Tower Bridge. You can't go wrong. The Tate Modern is also right there.

Hilton London Bankside is nice. I'venot stayed there but I've seen the lobby/pool. It's very central. I see $163 on hotels.com for August 1st.

Citizen M Bankside is nearby and also nice. I've seen the rooms and they are very "interesting" - you can press a button on a remote and turn all the lights red for example. I see $177 for August 11 on hotels.com.

To find the cheapest just look for the Travelodge's (which is cheaper than Premier Inn by a little bit) or an older boutique hotel that might be having a sale. There are one each in Bankside.

Try Motel 6. Even if your flight or taxi are delayed … they’ll leave the light on!