I've copied in a reply to a request for suggestions based in West Sussex and Hampshire. I know you haven't included West Sussex in your itinerary so parts of if may not interest you but it's a rough guide:
I would start my visit in Chichester, https://www.visitchichester.org. It's a very attractive City and very historic. There's much to do and see in the city and surrounding area plus there's a good choice of accommodation.
I would include Petworth House in your visit, https://www.nationaltrust.org.uk/petworth-house-and-park. It's not far from Chichester and ticks some of your boxes.
Petersfield is another nearby old market town with Queen Elizabeth Country Park for some gentle hiking. http://www.visitpetersfield.com https://www.hants.gov.uk/thingstodo/countryparks/qecp
There's a hike I enjoy (and the kids do too) and that's up Old Winchester Hill https://www.nationaltrail.co.uk/south-downs-way/attractions/old-winchester-hill. It's not too strenuous although depending on what side you approach there is quite a steep incline for 100 metres or so as you reach the top. I always like to coincide this walk with a visit to The Shoe Inn in Exton for a well deserved lunch after http://www.theshoeexton.co.uk.
Winchester should definitely be on your list. England's ancient capital https://www.visitwinchester.co.uk. A wonderful city with much to see and do, I just love the fact that King Arthur's Round Table is hanging in The Great Hall https://www.visitwinchester.co.uk/listing/the-great-hall/.
Your drive between these places will take in many fantastic small villages full of thatched cottages, meandering streams, cosy old pubs, village halls and a variety of markets. In fact the whole area that covers the places I've outlined is so full of such villages that it is impossible to list them all or recommend any in particular.
If you do have any extra time I would add in a visit to the New Forest National Park https://www.thenewforest.co.uk.
http://www.westsussex.info is a website that is full of useful information likewise https://www.visit-hampshire.co.uk.