I love the architecture of the Barbican - it is so important to its historical context, at a time when London was being shaped for modernity after the war - they also aspired to make it a bold art centre. We need bold, always.
The Design Museum is a museum in Kensington, next to Holland Park, which covers product, industrial, graphic, fashion and architectural design. There is also a nice cafe on the 1st floor with beautiful views of the park.