Being a newbie to this post & not having read the other 27 pages what we're looking at is the rear of a "T" shaped building with a separate building beyond (as it has a different style chimney) running at right angles to the main building (if it sits on a pitched roof that is) & at the other end another (looks like newer) 2 storey building with a large greenhouse in between.
2 observations, 1, there are park style benches which indicates (to me anyway) not a private residence. 2, all of the upstairs windows are open. May be just coincidence but looks like a convalescence home for TB sufferers.
I do know that during WW1 many gas casualties were sent to the Channel Isles to recuperate.
Just a thought.
Someone mentioned the pipe running through the tower roof. Stench pipe from the loos.