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British History Online 1 gives a full history of this house which in the Tudor period was also known as King's House or King's Place. 

British History Online 2 adds that Henry Algernon Percy, 6th Earl of Northumberland died at this house in 1537, and also explains the Brooke House name: "The Rt. Hon. the Countess Elizabeth of Oxford was the daughter of James Trentham, of Rowcester, Staffs., and at one time was maid of honour to Queen Elizabeth. She was second wife to Edward de Vere, .... The Countess became tenant of the mansion ... in 1596, and retained it till 1609, when she alienated it {transferred ownership} to Fulke Greville, Lord Brooke."

We arrived at the gates and could see people inside the building. For a good close-up photo of the plaque we needed to get through the gates. Pressing the button on the access keypad called forth a repeating message, in an American accent: "The person you have dialled is not available now, please dial later. The person you have dialled ..."  Annoying if you are using a phone; infuriating if you are standing in the street.  Eventually a delivery driver got the gates opened and we piggy-backed in.