The Barbican is worth visiting for it's brutalist architecture alone. The concert programming is really diverse; it's home to the BBC Syphony Orchestra, but you'll also find bands like the Dirty Projectors and the National amongst the classical and contemporary composer line up. Plenty of art, dance and theatre too, however, the Conservatory is the real gem that's not to be missed.
I grew up in Brighton which is full of cheesy bars like this. If you can handle being surrounded by multiple screens playing Kylie at full volume, its pretty much the cheapest bar in central london. The cliental is so varied its great for people watching and completely un pretentious.
Paul Rothe n Son has been there 120 years, bang in the middle of town. Hard to find a proper old-school deli in central London nowadays, n none more proper than this place. Great jacket potatoes, soup, sandwiches. It looks the part too. And the service is always great. Try and avoid the lunch queue.