Well, to do all those things in London, I think you'll want to stay closer to the city trouristy areas...to maximize your time at these attractions. Otherwise you'll just be sitting in traffic trying to get into and out of london!
It may not be the type of lodging you want, but if you stay in the picadilly circus area (read: tourist area) those things are only a quick tube ride away.
Itinerary looks good too...I hope your folks are in good health! Just reading it made me tired! jk
Aww man...now I miss England again.