Is hiring a car completely out of the question, as this schedule is virtually impossible by public transport and it will be difficult to find a taxi in many places. Sunday often means engineering work on the rail lines, so google before you travel.
Check out the websites of the gardens you plan on visiting, as all have a how to get there section.
Rye to Northiam (Great Dixter) - almost impossible by bus on a Sunday. It's less than 10 miles but is likely to cost more than £20 by taxi. Say 2 hours to look round the garden and house, minimum. Dixter is up a long rural road and there aren't any taxis nearby. You would have to prearrange or telephone for a taxi when you are ready to leave.
What are you going to do with your luggage? These gardens don't have a left luggage facility.
Northiam to Sissinghurst - you would need to get a taxi to Robertsbridge or Rye station to then catch a train to Staplehurst, which is 5 miles from Sissinghurst. It would take over an hour plus the service only runs hourly. Easier just to take a taxi right through - cost guess would be c £30. You need say 2.5 hours here minimum plus time for lunch.
Taxi to Staplehurst. As you want to visit Knowle, stay in Sevenoaks, not Tunbridge Wells - it's just a couple of stops further up the line and means that you won't have to fight for a seat on a train on Monday morning with commuters with your luggage. IMO, Knowle gardens aren't as interesting as the others on your list, as they are a parkland setting and deer park. Much of the interior of Knowle is closed his year for conservation work to be undertaken and what is open isn't on Mondays! The park is open. I would consider dropping Knowle if I were you. Other garden options nearby - Groombridge Place or Penshurst Place, both near Tunbridge Wells. Pashley Manor is good, but closed Mondays.
Hever or Leeds Castle would be a full day.
Chartwell - difficult to access by public transport. More interesting for its former occupant, rather than the house and gardens being outstanding in their own right. Needs c 2 hours.
Wakehurst - not the most exciting NT garden IMO. The nearest rail station is 6 miles away and it maybe easier going via London, as it is in a different line to Sevenoaks or Tunbridge Wells.
Nymans - no public transport nearby, so take a taxi from Wakehurst. If you are flying back to the States on the Tuesday, you will need to be at Gatwick at 10am, so no time for anything this morning.
Many people open their gardens for charity at weekends and these offer something different than the very staid (and slightly boring NT gardens and yes - I am a member!) - check out ngs.org.uk for the time you are visiting.
Trying to get a series of taxis from some of these gardens is going to be more stressful than driving! You also have the luggage issue. Driving makes so much more sense! Sorry.
