I tend to stay in the Ibis London City when going on my own dime. 10 minute walk to tower bridge, directly next to Aldgate east tube station.
It isn't as cheap as other Ibis hotels but it is very central indeed.
London Hotels (or even B&B's) are ridiculous expensive generally speaking, so doing a search on the usual sites like hotels.com, trivago, etc. is always worth the hassle as you might find something on offer that suits you.
My experience with B&B's or small Hotels close to London City hasn't been the best though, there is some shoddy places around.
HTH.