I assume this is Coney Hall on this 1907 map (shows "Coney Hall Farm" and "Coney Hill" a little north.
https://maps.nls.uk/view/103676678A few mentions of Coney Hall in newspapers, including a number of letters to the editor regarding Sir John Lennard putting up a wall enclosing Wickham Common
(Saturday, May 10, 1890, Publication: Daily News)
Modified to add:
Still being advertised in 1959 - the Shaftsbury Society at Coney Hill, West Wickham, though I guess it may not be in the original building?
Wednesday, Mar. 4, 1959, Publication: The Times