A summer in London with your 6 year old - you are going to have an AMAZING time! There is so much to do here aimed at exactly that age group.
I can understand you wanting to set up a social circle for your daughter since you will be here for a while. I think your best bet will be through your wife's work: colleagues may have young families, or have friends who do, and be happy to meet up for outings. I wouldn't get too hung up on them being gifted. Most middle class London families enjoy all the cultural enrichment we have available, and I'm sure your daughter will enjoy being with other kids regardless of giftedness.
You could sign your daughter up for some holiday activities and again be very forward with other parents you meet about being interested in meeting up, but people are often already quite busy. Have a look at these
But you can do a lot of fun things just with your daughter. You have rightly spotted that the museums do a lot of activities and workshops for children. As well as the ones you mention, also look at the London Transport Museum, Science museum, British Museum (mummies!), Museum of London, Museum of Childhood. Be aware that the big museums get REALLY busy during weekends and holidays, especially the Natural History Museum. If you are here during term-time and your daughter isn't in school then take full advantage of being able to go during school days.
Some of the museums (eg Science Museum, British Museum) do overnight sleepovers (you bring sleeping bags and sleep in the museum! Lots of activities before sleeping). You have to book well in advance, so you should look at that now.
There are some great cultural activities for children in London:
- LSO, LPO, Royal Albert Hall and Barbican do classical music concerts aimed at children (you'll find more music events for children if you google).
- There are some really outstanding childrens theatre companies: look at the Little Lyric Theatre, Unicorn Theatre (London Bridge), Little Angels Puppet Theatre (Islington).
- There is often open air theatre in summer, so search for that too (the amazing Regents Park Open Air theatre doesn't do so much for children these days, but there are other smaller ones).
- And so many west end shows and musicals to choose from too!
There are some fab parks and open spaces. Check out the Diana Memorial playground in Kensington Gardens,Coram Fields, Battersea Park Zoo (also has a huge playground).
Of course, you'll have to visit tourist attractions too.
If your DD likes the Tower of London and Buckingham Palace, then go to Hampton Court too. Do the journey there by boat along the Thames one way - the Palace was designed to be approached by boat and is very impressive from the river (train back is quicker).
One you might not think of is the Houses of Parliament: the guided tour at weekends and during the summer is well worth doing, or you can watch a debate from the public gallery.
Worth going out to Greenwich for the fantastic museums there and the Cutty Sark. You might like to go on the cable car too.
London Eye is fun. Southbank and the aquarium, changing of the guards, cabinet war rooms, lots of markets and interesting places close to Shoreditch. I like the London Museum of Water and Steam, although that would be a reasonable journey for you (it's near Kew). In fact, there is a huge amount to see if you are willing to travel an hour or so - which I would recommend doing if you are from far away. Too many great things to mention! Have a wonderful time.