Bluntly, OP, this sounds like more of a 'you' thing than a London thing. If you're afraid to be out in the early hours but your friends aren't interested in doing what you do, don't like being out in cloudy weather, haven't been clubbing since university, can't figure out how to get home from central London to Mile End by night bus, and usually just go home after work apart from once a week -- then frankly, you may as well be living in some dullsville rather than in a great world city.
It's a while since I stopped living in London (left the UK), but when I did live there, I did things like getting cheap theatre and opera tickets, going to all the markets (always started Sunday morning with Columbia Road and then down Brick Lane to Spitalfields for food), I was a member of the Tate and Royal Academy, so did members' late night events, openings etc, always went to the London Film Festival, music at the Wigmore Hall, free Radio 3 lunchtime concerts in various churches in central London, Open House, one of the singalong films at the Prince Charles, taken classes in fencing, philosophy, breadmaking and drama, did a lot of walks using books like 'Village London' and 'Secret London', and looked up and visited a lot of houses where famous writers lived. Lots of events at the LRB. We did a lot of charging off to fairly obscure places for food (like the best ever chilli paneer in Wembley) or historic pubs etc. DH went to a lot of football games. I used to swim in the Hampstead Ponds and go to a fair few gigs.
If I were in London right now this minute, I would be booking the current Punchdrunk show out in Woolwich, see The Bacchae at the NT and the Emily Kam Kngwarray show at Tate Modern and Gilbert and George at the Hayward, have tapas at Moro, have a walk for old time's sake on Hampstead Heath and a few drinks in The Flask, and eat my bodyweight in good Indian food, which is sadly lacking where I live now.