Nothing says London like iconic Parliament Square.
Agree walking is the best way to see many of the sights as many are closer together than you think.
ALL within walking distance:
Walk up Birdcage walk into St James Park up to Buckingham Palace. (please don't waste precious time on trying to see the Changing of the Guards)
Instead use your short amount of time visiting any of these Hyde Park, Kensington Gardens, Trafalgar Square, Covent Garden.
See Tower Bridge, Tower of London and then walk up to the glass covered Leadenhall Market place and then a long but pleasurable walk along the Thames to the Blackfriar pub across the Road from the Blackfriar Tube stop and from there up to St Paul's.
Use Google maps satellite view to see how close things are and if you buy a travel card you can use the tube and busses to help tired feet.
Lastly if I were to see one interior that epitomizes London I'd spend some money and take the Verger's tour of Westminster Abbey. Utterly fascinating!
Enjoy your 48 hours. You won't be bored!