Well there's the Premier Travel Inn at County Hall.....although I notice the prices have gone up quite a bit since we were there last year.

It's right next to the London Eye, so you can walk along the South Bank to the Globe, and also get a tube to the West End, or walk back via Trafalgar Square and Hungerford Bridge

We never bother with the breakfast there, there is a little shop opposite that's open all hours, or of course you're never far from an eatery in London.....

There's also a Premier Travel Inn at Southwark. If you know the area it's immediately behind (or part of?) The Anchor, Bankside which is very close to the Globe. Lot less walk than from County Hall but further from the a tube station - Southwark/Borough/London Bridge ?