I spent a lot of time in the British Museum whilst studying for my Degree. My work has changed considerably since then but it is still a place I return to again and again. Inspiration doesn't always come from objects in the collection but also from the space and its visitors.
A walled secret garden, the Physic Garden is like a time capsule, founded in 1673 to allow apprentices of the Worshipful Society of Apothecaries of London to study the healing properties of plants, it is now a living museum. Tours are offered by volunteers with immense knowledge of botany and the cultural history of the garden or visitors can wander the grounds alone and spot odd and fascinating plants such as the mystical Mandrake.
A great place to start the day if you are in South London is Federation Coffee in Brixton Village. The food is great and so is the coffee. This is a very chilled place where you can get both and indoor and outdoor experience (even though the whole place is under the Village roof).
A small place to eat with great vegetarian/vegan food just off Camden High Street. This place does an amazing buffet at a great price, as well as serving reasonably priced tea and juices.