For the markets, I would say Greenwich and Borough markets are a good bet - you will pass Greenwich market anyway, and Borough market is a short (and picturesque) walk from the Tate Modern (open Fridays and Saturdays only, though).

Otherwise:

The Leicester Square area isn't particularly pretty or attractive. While there's plenty famous sights to see round there, it's not somewhere I'd recommend lingering - it's tacky and overpriced. However, nearby you do have the likes of Foyles (excellent bookshop with jazz café), the Photographers' Gallery (again, excellent café, very interesting work), and the back streets of Covent Garden to wander round. The area you're describing is compact enough to render a hop-on-hop-off bus tour rather unnecessary - these are better for getting a flavour of a wider area of London.

Around Trafalgar Square, you've also got the ICA, with a very good bar and some interesting exhibits.

For the Sunday and Hampton Court, you're ideally placed to get there, as there's a direct train from Waterloo. However... Be careful about engineering works and check http://www.southwesttrains.co.uk/SWTrains before you go. You'll be able to get there, but the timings may be different, you may have to change, etc, etc. You can also buy train/admission packages from SWT, and do try the maze!

There's a very scheduled good river boat service now, running every 15 minutes, and, although you'll have been on the cruise, it's still a good way to get to Greenwich - and you can get from Tower Bridge Pier direct to Greenwich pier on it. Otherwise, there is a train direct from Greenwich to Waterloo East every half hour, but the return via the DLR (between the skyscrapers of the Docklands) is actually rather fun, particularly if you get a seat at the front.

Harrods is pretty poor really, although the kitsch of the Diana & Dodi memorial almost makes it worthwhile...

For getting around, don't forget that London is quite a walkable city, and that your hotel is very well located. The tube map is far from a physical representation...

For example, from your hotel to Trafalgar Square, via the Jubilee Bridge, is less than a 10 minute walk. Trafalgar Square to Leicester Square is about 5. The London Eye is 5 minutes from your hotel, about 10 minutes from the Tate, which is again about 10 minutes from Borough and only 5 from St Paul's.

Personally, in Central London, I prefer to take the bus for longer journeys rather than the Tube - it's more pleasant, and you see more. They're also running 'heritage' routemaster buses on some key tourist routes (like the 15, which includes the stretch from Trafalgar Square to the Tower) which is a proper London experience in itself.