Canterbury
Dover
Leeds castle
Hever Castle
Portsmouth (Mary Rose & HMS victory)
Arundel--for the castle
Bodiam castle
Salisbury--Don't miss the cathedral and trip out to Stonehenge
Winchester
Hampton Court Palace--easily done from London by train
Have a look at this thread:
https://community.ricksteves.com/travel-forum/england/southern-england-and-london-after-landing-in-gatwick
It covers a lot of great sights and there is a good discussion here.
Take into consideration that England has a great train system and that you will be able to use the trains to get to many destinations. Do not automatically assume you will need a car. Plot your destinations first, then figure out how many you can get to by train.
For example, Salisbury and Bath (both popular destinations) are easily reached from London by train.
Brighton, on the beach of the south coast, is easily reached by train from London.
Canterbury and Dover--fast train from London will get you there.