Maybe an obvious choice, but the Barbican is always good. My favourite space in there is the curve gallery, last time I went was for Cory Arcangel's 'Beat the Champ'. I love the architecture, although not many people do. I think it was voted 'London's ugliest building' in 2003.
London is blessed with many fine parks, but Brockwell is a real jewel in south London. A short walk or bus ride from Brixton, or right by Herne Hill overground station, the park includes large grassy areas, the fantastic Brockwell Lido (a vast open-air pool – hectic mid-summer unless you get in early - a 7am swim is utterly blissful), tennis courts, a bowling green, a BMX track, a mini-railway, a secret garden, and a few newly landscaped ponds. Perfect for a lazy summer day.
The best place for a glass of wine*. It's the perfect balance – not too formal, not too informal either. Food is consistently good. Lamb's Conduit Street is a little oasis in the midst of Bloomsbury. (*slightly overpriced)
Free natural ponds, open all year round (check online for varying opening hours according to seasons). Get an absolute get away feeling right in the heart of London, a treat if you enjoy swimming with ducks. Amazing park to enjoy a few games of cards and a walk around afterward too...