You can't move for National Trust houses & gardens down here.
A few worth specific mention are: Sissinghurst - home to Harold Nicolson & Vita Sackville-West and a favourite amongst garden lovers. (30 minute drive) See: www.nationaltrust.org.uk

Then there's Great Dixter, by Christopher Lloyd who was one of Britain's most talented plantsmen & writers. that's just 10 minutes away. See: www.greatdixter.co.uk

Small Hythe Place, 16th century house and home to many - most famously the Victorian actress Ellen Terry is 10 minutes away. See: www.nationaltrust.org.uk