Make sure you catch a real
basement jazz gig at the
606 in the backwaters of Chelsea area.
http://www.606club.co.uk/
It is a jazz institution. You'll have to eat there and pay a cover charge if you are not a member. Food just OK, but the intimate atmosphere and groups are very excellent. You will need to make reservations over the phone in advance.
Not necessary to go to Ronnie Scotts which has lost alot of its soul since the new owner took over 3 years ago and swished up the place from its much-loved authentic salubrious late 60s/early 70s decorum, which in the pecking order of cool jazz interior design for London jazz clubs has been handed over to the 606!!!
Also a trip to Kew Gardens cool.
http://www.kew.org/ Great little Kew and its Station Parade. Tube gets you there. Richmond last stop on that tube line and also very neat on the river. You might even run into loud mouth and leggy Texan Jerry Hall

or see some old rockers running about! Great view up the Richmond Hill Road, which is elevated and looks to the west, right next to Micks' (and Jerry's) and Pete Townsends' places.
Also Tate Modern, but get ready for lots of tourists...its such a success.