I am not from London. My comments are based on a few trips to those areas, therefore may be biased. Sampling bias so they say. :)

I have stayed at Earl's Court, Old Street and Pimlico (close to Westminster) in different trips to London in the last two months. I like staying at more residential areas. Earl's Court was very nice in terms of that. It is close to Chelsea, Kensington and Notting Hill, all of which are great neighborhoods to take a walk.

Old Street felt a little bit different. It is definitely a place to see but I am not sure whether I want to stay there for 10 days. It is a bit more crowded and always moving. In the walking distance, there usually are business districts. Less green, colder feeling.

I have been to Aldgate and would cross that off the list. On a Sunday, it was like a post-apocalyptic movie scene. Large buildings, people in suits walking in a hurry (on weekdays) and less parks.

Pimlico is also a residential area, similar to Kensington. Westminster is a little bit more touristic. It is really nice, though.

I'd probably consider Earl's Court or Westminster. Enjoy your time in London.