You could fit loads in on a (9 hour-ish) daytrip BUT it would be very tiring! The best thing to do would be to 'contain' your trip within some geographical parameters.
Your DD wants to see Buckingham Palace. Do your other DCs have any preferences to work around too? That's your starting point
You could walk from BP down the Mall to Trafalgar Square (thro' Admiralty Arch), do the National Gallery, walk down Whitehall, taking in Horse Guards Parade to see 10 Downing Street, Houses of Parliament (and Big Ben) and Westminster Abbey then cross the Thames and walk along the Southbank: a trip on London Eye, visit to the London Acqarium, Shrek Adventure (London Assembly building) - if you're feeling flush, Globe Theatre, Tate Modern ending up with a walk thro' Borough (foodie heaven) market...along to HMSBelfast with great tour of Tower of London and Tower Bridge... (you could also get Routemaster Bus No. 15 which will give you a great ride
But there are loads of other options using Buckingham Palace as your starting point. You could walk up to Marble Arch from there, get a bus along Kensington Gore, get off at Royal Albert Hall, do Science/Natural History/V&A museums, walk along to Harrods and/or walk thro' Hyde Park, Kensington Gardens (including a play the Diana Memorial Garden), get a bus...
You could could do some of your sightseeing on buses which give you a birdseye view of London - either the tourist variety, or if you plan ahead you can also use normal London buses. For example, there are still Routemasters on the No 15 route taking in loads of tourist London (including St Pauls, Fleet Streetfrom/to Charing Cross Station (near Trafalgar Square).
These are all mainstream touristy things to do, so let us know if you are looking for more quirky options.