A place that really grows on you, with each visit the Barbican reveals more of itself to you and your bond is tightened. The Barbican Kitchen is great for coffee or lunch, summer hangouts by the ponds, cinema in winter, the amazing Barbican Hall for live music, and don't miss the conservatory.
The V and A houses a public collection of prints and drawings of every kind imagineable from old master drawings to photographs, playing cards, fashion plates and wallpapers. You have to select what you want to see from the collection http://collections.vam.ac.uk/and book in advance. The behind-the-scenes route up to the study room of v and a is a treat alone (pictured)
Founded in 1828 the museum houses around 67,000 specimens many of which are now extinct or endangered. My favorite is the skull of a Bottlenose Whale from 1860.
On a hot day there is nothing better to do in London then go for a swim in the Heath ponds. You get completely transported you out of the city as Hampstead feels like being in the countryside except you're in central London?!