We owned at Broome Park and loved the three times we went there. We took the Eurostar from Ashford station and went to Paris for a day. Long day but worth it. If you go, plan to use the Batobus which takes you along the river to the major stops and is much faster than trying to get around by cab. Very important for a day trip and very relaxing ride.
Having said that, in Kent we also went to Leeds Castle, Canterbury, Goodnestone Gardens, www.goodnestoneparkgardens.co.uk
, Chartwell (Churchill's home), and Dover Castle, which has an underground hospital from WWII. Very interesting. You could also visit the grounds of Battle of Hastings (found in the town of Battle). Each of these would be day trip although Canterbury is fairly close by.
I think those are the major places we went. Let me know if you have more questions.