I would heartily recommend the
Natural History Museum - quite apart from all the exhibits the old part of the building is absolutely beautiful - and entry is free. It is far and away my favourite attraction in London... and I've been to most of them.
I know you say you don't want to do the usual tourist things but the Tower is very interesting & Kew gardens is lovely on a nice day. For a lot of the other sights some kind of walking tour is good. I suppose all this depends on your interests - if you are into Shakespeare (is that why you are heading to Stratford-Upon-Avon?) then the
Globe theatre is going to be a good option.
The Tower Bridge Experience is good too - you get to go up to the top walkways of the bridge and there is a small museum attached - could easily form part of a walking tour.
Depending on how many attractions you do want to visit a
London Pass could be good value for you. There are options for different numbers of days and also to include free public transport.
I don't thing Stone Henge is worth a full day but someone else mentioned teaming it with Bath which is a great idea - The Roman Baths exhibit is great andit is generally a very pretty city.
Windsor is also nice - and has the castle.
Hope that helped - enjoy your trip!