Few suggestions:
Get the RS GUIDEBOOK and read it. Will truly make your first trip there much easier. Practical information about sites, accommodations, London Underground,etc.
Make a list of what’s important for you and your sister to see. Selfriges and/or Tower of London? Theatre?
St Paul’s and/or Carnaby Street? Museum of London and/or Portobello Market?
Take a look at the London Underground tube map and understand what you have expressed seeing and doing are all in Zones 1&2.
Use Google Maps satellite view and type in Big Ben.
Then type in Buckingham Palace. See how close they are. Very walkable city.
Lastly it’s an expensive city. Very important for you
to understand that when you start looking for accommodations. You will also need to understand hotel rooms are small. Otherwise you’ll be totally surprised.
Personally I’d go for the Premiere Inn near the London Eye. Remember you are traveling at the height of tourist season so crowds will be huge and you need to book accommodations now.
Have an enjoyable first trip. Great city!