Hi Samantha

How close to the action do you want to be?! If you are looking to be pretty central I would suggest Little Venice  (Warwick Avenue tube), it has some lovely canals, is very central but doesnt feel too manic. Also consider Battersea Park (Prince of Wales drive), the park is brilliant and is extremely close to Chelsea. Putney and Richmond  are also worth a look but a little further out, there is more open space, lots going on by the river but still close enough to London to still enjoy its benefits. As to what you should do, that is a longer answer but needs more info from you first in terms of what you are interested in and what your budget is!

Good luck

Nick

All good ideas here.  Consider the Hop-on, Hop Off Big Red Bus, this gives you the lay of the land; then you can go back to places of interest.  Maybe go to 222-B Baker Street to the The Sherlock Holmes Museum.  Besides being fun, it may encourage your kids to read (Sherlock stories are always fun).

An Airbnb in a nice, fairly central neighbourhood is a good bet with kids - somewhere in Zone 2 e.g. round Queen's Park (great little park & high street) or West Hampstead are examples near where I live, it's easy getting just about anywhere from here as the transport links are excellent (make sure you download Citymapper on your phone if you haven't already got it, it will allow you to travel like a Londoner) and yet you're out of the exhausting and expensive central tourist traps. 

Go to: the London Zoo, the London Eye on Southbank, Hamleys toy shop on Regent Street, the Science Museum, the Warner Bros. Studio Tour London in Watford (about an hour out of London but really worth it if the kids are into Harry Potter). Catch the river boat from central London up to Greenwich and have a picnic there, visit the Cutty Sark, see the Greenwich meridian. Madame Tussauds London has horrendous queues, but kids do seem to like it! Regent's Park has a boating lake and is a good spot for a picnic if sunny. 

Don't make the mistake of staying anywhere near the airports. The transport links from the hotels out there into the city are slow slow slow and you will end up wishing you had stayed more central.

Aim to find a hotel in Zone 1 or at a push Zone 2 on the London Underground and you won't end up spending all your time traveling.

I'd consider an AirBnB for a place to stay; its' nice to have a small kitchen and possibly a washing machine with kids. Depends where you are going and what you can afford.

As for places to go my boys loved the Museum of Science right next to the Natural History Museumnear Hyde Park. That is a whole day by itself. Another hit was the Ceremony of the Keys at the Tower Of London though that is at night...a rest before could be helpful.