For London, we've used an apartment rental service called Ivy Lettings. We were usually traveling with our 2 daughters, so we got more space, a kitchen and 2 bedrooms for about the same as a 2 rooms in a nice hotel I know VRBO and other services have lots of choices, but for overseas rentals, we've preferred working via a rental agency that's been in business for a while and has vetted the places they list. We also know they will be available if there's a problem, vs. the individual renting an apartment directly. We look for a place within a couple of blocks of a Tube stop and in a neighborhood with grocery stores and restaurants. After staying in timeshares, we really like having the space to relax and eat some meals in, beyond just a hotel room.
In advance of a trip like this, I usually review the Rick Steves book, because he does a good overview of the top sites in an area. Then I'll read through Frommers or Fodor, because they go much deeper and may have other sights that appeal to our personal interests, like a toy museum near our apartment when our kits were younger.
There's so much to do in London and surrounding areas, it really depends on your personal interests. We loved the London Eye, British Museum, Churchill War Rooms, Westminster Abbey, Tower of London, visiting some of the market areas like Burough Market, checking out the food displays in Harrods, and just walking around some of the different neighborhoods. We usually like to do 1 or 2 major sites in a day, then use the rest of the time to just take in the local color around the area. If you want to venture out from London, Bath is an interesting city and would also give you a chance to see Stonehenge.
I've always found the research and planning to be part of the fun of taking a trip somewhere new!