I second travellingfrog on taking the 15, especially if you can get an old routemaster! fyi though, you have to use oystercards on the old bus, they don't accept contactless cards (
) so I never take it anymore :-(
the 55 bus actually covers most of the same route as tourist buses. I've taken a couple of friends on that route when they're visiting.
Brick Lane is quite cool, and Fournier street, right close to it, is one of my favourite streets in the entire city. It's very old and has held refugees from the 1700s onwards. there's a gallery at one end (the opposite end of the mosque) with some really excellent old maps of the street; if you go in and ask the gallery owner they'll show you to the secret garden shed of old maps out back.
the ten bells pub is right around the corner, which is where jack the ripper found some of his victims. it's only slightly updated and quite enjoyable. and of course spitalfields market, across the street, has amazing food and interesting artisans selling their wares.
also a plug for the food market by goodge street tube station (i work a 5 minute walk away). all street food, some of the best lunches in london, for around a fiver. I especially like the peruvian chicken - it's to the right of the american church and all the way to the back, absolutely outstanding food.
if you end up in the starbucks across from the british museum and see a couple of nerd-types on laptops talking european regulations and data security...that may very well be me and my coworkers, stressed and offsite to try and wrestle some projects to completion :-D