You're not exactly in an ideal position for sightseeing given most places will be either closed or preparing to close when you get there - a 2pm arrival means you won't get checked into the Sofitel and back out of your hotel room until 3pm. You're then looking at about 75 minutes to the east side of London on the tube, and not too much quicker even if you shelled out for the Heathrow Express.

Most of the threads linked above deal with early arrivals and late departures, so they'll be of use to you more for the transport options - still worth a read though.

Looking at the things you want to do, I'd be inclined to take the Piccadilly Line tube from LHR T5, changing on to the District Line to Tower Hill. The Tower of London is directly outside the station, and Tower Bridge is adjacent. Walk over Tower Bridge on to the South Bank, and then along the river - you'll be at Westminster for the London Eye (which would still be open) in about 45 minutes, and you'll see St Paul's Cathedral (on the North Bank), the Tate Modern along the way and the Palace of Westminster (Houses of Parliament, where Big Ben is) on the other side of Westminster Bridge - it really is a pleasant, relaxed walk with a nice vibe and some decent views.

By the time you've had something to eat - plenty of pubs and restaurants in the area - it'll be heading towards 9pm, and you'll need around an hour or a little more to get back to the Sofitel.

In terms of rain, we don't really have a rainy season here, and there's a possibility of wet weather in every month. No-one is really going to be able to advise you accurately - you'd have to play it by ear on the day.