I would STRONGLY recommend doing one of the 'taster' tours (called different things at different places) at, for example, the V&A and the British Museum - even if you have been to the museum before. Years ago I lived in an awful bedsit in north london and my boyfriend lived in another city. I hated being in my flat for any time and got out of London whenever I could but whenever there was nothing else for me to do I would do a tour like this. I loved them all and learned so much - I'm a regular museum goer and generally do my own thing but these tours were amazing and pointed out things you'd just not see if wandering yourself - real hidden gems. Even if these museums aren't normally your thing I think you might enjoy.
I would also recommend doing a couple of walking tours like [[http://www.walks.com/ these]. I've done a handful over the last few years, including one in my own neighbourhood. It tends to amuse the guides if you are very local (and make them a bit self conscious) but again I have always enjoyed and learned new things. A couple of them are built on a pub crawl model - also a good way of finding a few good drinking establishments you may not have discovered before and you're with other people so no need to feel like you're drinking on your own!!
Another thing I would do if I had my 'free time' again is evensong at Westminster Abbey or St Pauls. I'm totally not religious but did both of these and they felt like incredible experiences and I think very very few 'Londoners' have tried it.
Just realised you said August and started this in June, so may have already completed this week off. Would be very interested to know what you did and how you found it!
Theatre, obviously. IMAX. Outdoor film screenings with some takeaway food. V&A (all the nooks and crannies that there isn't time for normally). Greenwich. Brick Lane curry. Colombia Road flower market. Forbidden Planet (possibly just me). Guildhall, Barbican, whatever's on at South Bank. Scoop at More London free festival. The thing about London is that even if you've "done" a place, there's always something new or special going on, especially over the summer.
If I had a week in London on my own this summer I would: go shopping on the Kings Road and/or have a day visiting my favourite shops in various parts of London, visit any/all the summer exhibitions at the various museums/art galleries that took my fancy (Pompeii & Herculaneum, Lowry, etc), have a day at the seaside (if it's Brighton, visit the Royal Pavilion), visit Hatchards, Daunt Books, etc, arrange to meet friends for lunch/drinks/dinner on a couple of days - even if it was just lunch near their work, walk up Parliament Hill or go for a walk on Hampstead Heath or Primrose Hill, visit Borough Market on Fri, spend Saturday morning wandering around Notting Hill market ... is that enough for you?
I'm going to have a week off in August in London and don't know what to do with myself.. I wanted to travel but now think I will stay in London as don't want to travel on my own. Thing is I've lived there for two years so have done a lot of things and are looking or more 'must do things' that are the typical touristy things... or places that are easy to get to on the train.. I'm thinking one day in Brighton will be on the list.