If you're coming into Paddington, don't take a taxi to South Kensington. You'll be stuck in traffic for ages. It's much quicker and easier to take the tube. A lot of non Londoners make the mistake of thinking getting a taxi will make their life easier, but most of the time it really doesn't unless you have a large family/lots of luggage. With just one DD to take care of, the tube will be a much better option - and she'll have the experience of it.
NHM during the summer holidays is going to be rammed. You can sometimes wait for an hour to get in. I would advise that you pre book and get there bang on opening time. You'll have to queue your way round the dinosaurs - so you'll be in there longer than you think.
I would suggest buying sandwiches in Pret at Paddington and taking them with you as the food in the museum is crap and overpriced - you can then walk up to Hyde Park after the museum and eat your sandwiches while enjoying looking at the Royal Albert Hall/Albert Memorial.
After lunch, get back on the tube at South Ken and get the Piccadilly Line to Covent Garden for your show. Again, much quicker to tube it than taxi.
After the show, you can jump in a taxi for the brief journey to Hamleys (if it's too far for DD to walk - I suspect it would be as it's a 15-20 min walk from Covent Garden to Regent Street) and spend some of the money you've saved for taxis on a treat for DD, then jump on the tube at Oxford Circus back to Paddington.
If you did want to do an afternoon tea at any point, on Regent Street there are a couple of branches of Caffe Concerto, which is a very nice place to stop for tea and cake and doesn't cost a fortune - if I remember rightly there's one pretty close to Hamleys.
I wouldn't try and fit in any more than this. If you're from outside London, it can be very overwhelming for a small child amongst all the crowds, and there's quite a lot of walking to do. When my nephews visit me in London I'm always surprised by how exhausted and stressed they get doing things that are just normal to me. It's nothing to me to walk for half an hour to get from one shop to another, or from a restaurant to somewhere else, but for them that's a huge walk and it's knackering. If your DD is used to mainly doing car journeys, you do have to factor the walking in!
Have fun!