What can I do with my pack?
Is it worth the time and money to travel into the city and try and see something or should I just try and sleep somewhere and save London for a real visit?
There are places that you can check your pack in, like lockers and other things...especially at main train stations. You can take teh train straight into London. Covent Gardens is a good location to go into as is Piccadilly Circus. The train, when I took it was about 45 minutes in, so allow for extra time. almost 10 years ago
You could jump on a train or tube and head up into London - depends on how tired you are going to be. London requires some level of energy, even if its only walking up and down stairs in stations etc.
If you went to Piccadilly Square you would certainly be in a busy part of London but also within easy walking of fairly quiet parks such as Hyde Park or St James's. You could get off a couple of stops earlier and head to the Natural History Museum or the V&A.
Covent Garden and Soho are also good areas for food, drinking and people watching.
London is good for just wandering around in - lots of back streets with little gems of cafes where you could sit and people watch - I like the back of Harrods or the back of Selfridges. If you were happy to travel a bit more then Camden and Islington are good or you could hop on the bus at the airport (111) and go down to Richmond or Hampton Court.
Dont forget though getting out and back into the airport will take you probably an hour or so each bit plus your travel into central London - you would end up with realistically around 6 or 7 hours to see London. almost 10 years ago