Hmmm...no obvious London things. Depends on what interests you, but here is a short list:
St. Stephen Walbrook -imo, the best of Wren's city churches - central plan, saucer dome, much enhanced by the Henry Moore altar added in the last couple of decades
David Linley - in Pimlico and now in Mayfair - expensive, but worth it to see what a top craftsman and designer can do - in 200 years people will pay a fortune at Sotheby's and Christies for this furniture. Plus you can overhear the staff on the phone saying things like "Sadly we're just back from Joburg" which is of at least some sociological interest.
Sissinghurst - hop on the train, see Vita and Harold's stunning garden
Diana & Dodi Memorial at Harrods - the campiest thing you'll ever see
Spencer House - one of the best, and one of the last remaining 18th century aristocratic town palaces - it's only open on Sundays, I think
Bibendium - great deco building, the food was still good the last time I was there, and there is a Conran shop at the back
The Old Bell pub in Ludgate circus - good City crowd, nice pub, stand outside the back door with your pint and admire St. Brides
Just wander around aimlessly - there's something interesting in London on almost every corner.
