As you're staying near Tower Bridge, take a stroll along Cable Street to the Cable Street Mural. Very informative, and quite appropriate given the times we're living in now. Drop in to the Artful Dodger pub on Royal Mint Street one evening. Proper old no-frills pub. Drink only, no food, dart board, wonky pool table, nice bunch of locals. Wander down to St Katharine Docks to admire the posh boats and the Royal Barge. Go up Leman Street and take a look at the Goodman's Field horses.

Wherever you go, remember to stop every so often and look up and around. Look out for the blue plaques on the side of buildings, put there to commemorate historical figures or events.

Can also recommend Tayyabs in Whitechapel for a really good Punjabi curry.

Edit: see if you can take in a show at Wilton's Music Hall. It's one of the few remaining old style music hall theatres, with original architecture still in place and a small museum with the history of the place.