Do not stay too far out of the city centre, even if it is only a 20 minute train journey from the hotel to the city centre.
The problems come when you go into the centre of London for the day, but then want to go back to the hotel to freshen up and change, and then want to go to a show in the West End in the evening.
If you stay in, or near, central London you can pop back to your hotel to drop off shopping, or pop back for an afternoon nap, all of which you cant do if you have a longish train journey back to your hotel.
Hotels in central London are expensive and you are looking at over £100 a night as a starter for most hotels.
Note there is a dedicated London forum on Trip Advisor where they have travel experts who gave detailed advice on hotels, things to see etc. (not that you have not had some great advice already from AnswerBank).
There is a huge amount to see in London, and it can be difficult to narrow it down to what you want to visit.
I always suggest a day walking round the Westminster / Whitehall area is a good place to start for first time visitors as this includes the Houses of Parliament, Big Ben, Westminster Abbey, 10 Downing Street, Cenotaph, Buckingham Palace, St James Park, Trafalgar square, Churchill War Rooms, and just over the river, the London Eye. All these are enclosed in a fairly small area (and more besides).
Another day can be spent on the East side of London seeing the Tower of London (get there early and allow about 4 hours for this) as well as Tower Bridge, and the Monument (Great Fire of London) and St Pauls are nearby.
A walk along the South Bank of the Thames from Tower Bridge to Westminster Bridge (or vice versa) is also another way to spend a full day. Loads to see all along the South Bank (London Eye, Shakespeare globe theatre, Tate modern, Golden Hind ship, Millenium Bridge, HMS Belfast etc)
If you need more info just ask.