Sights? I think you have covered the key ones but are there others things you like? Museums, royal history, architecture? Maybe the London Eye? Personally I would recommend a decent guidebook, a map and lots of walking with a couple of bus rides to access St Paul's and the Tower.

There are not currently any direct London-Amsterdam trains though Eurostar are planning them for some time on 2016 but no indication exactly when. The change at Brussels is very straightforward between Eurostar and Thalys (the high speed service linking Paris, Brussels, Amsterdam). St Pancras is the London station and it is very close to the centre of London, stunning building. Book well in advance as you are travelling at super peak tourist period overlapping a key UK public holiday, everything will be open but very, very, very busy.

If ou really want to see some countryside you need a day trip from London which would be overkill on a 2 day visit. Save the train ride to somewhere like Bath, Arundel, Cambridge for another visit.