I worked at the Barbican for several years and during that time became very fond of its architecture. One of my favourite spaces within the centre is the conservatory, a giant tropical garden in the middle of a concrete city.
The Premises is cozy café attached to a recording studio in Hackney (frequented by Nina Simone, Al Green and Jarvis Cocker over the years.) Lovely people serve up tasty breakfasts and Turkish food in the evening.
The Almeida, in Islington, is a small theatre with big influence. It's an intimate place to discover groundbreaking theatre and get up close to some of Britain's leading actors. The cafe/bar across has a constantly-changing tasty menu, relaxed atmosphere and free wifi. Open all day and great place to spend time even if you're not going to catch a production.
If you want a peaceful day out of the busy city, Highgate doesn't really feel like London. The cemetery is interesting and peaceful, there's a tour of the old half which is definitely worth the money. Karl Marx, among others are buried there.
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.