david,
It was certainly there in October 1892, as Miss L M Ennals was summoned for allowing an unmuzzled dog on the highway and find 10 shillings, being reported in the York Herald.
There was a another ladies schools in Scarborough, at Clifton Villa, Ramshill Road, Scarborough, an advertisement in 1866 reads: Education by the SEA-SIDE-Clifton Villa, South Cliff, Scarborough, Mrs. Maynard from the South of England receives a select number of young ladies to board and educate. They will be carefully, morally, and intellectually trained, enjoying at the same time the comforts of a cheerful home combined with the mild discipline of a well-regulated school.
By 1872, this school had moved to 2 Rawdon Villas, Ramshill Road-Clifton Villa was sold to Edward Dixon, my great, great, great grandfather a retired fodder dealer for the army.
Have you tried any contact with Scarborough library, they have a good local history section and have helped me in the past.
I notice that a few pages back in the same census is an Antoine Wachter, principal of Ladies College, at 4, West Street. In the same directory of 1890 is this entry: Wachter Mademoiselles, French ladies' school, Gordon house, West street.
In 1896, Gordon House, 4, West St, was sold for £1,050 see The Yorkshire Herald, and The York Herald (York, England), Monday, October 12, 1896; pg. 3; Issue 14150. from 19th Century British Library Newspapers: Part II at your local library on your library ticket.
Another entry for that same directory: see
http://www.ramsdale.org/scarboro.htm, states the following:
Among the private establishments for educational purposes may be mentioned:- Oliver's Mount School, The Westlands, Westwood Collegiate School, The Uplands, Haddo School, Gordon House, Grammar School, Belgrave School, Grosvenor Crescent, School of Art, Clifton House, St. Martin's School, &c.
The Westlands, Westwood Collegiate School, Haddo School, did Cambridge University Public examinations & College of Preceptors- an examination with two objects in view, to gain a diploma, and also a way of having a common examination for different schools in the locality.
So maybe by 1896 the school had moved on somewhere else, or try the local library and see what they can do for you.
Tom