The proprietor of the asylum George Robinson wrote a book
George Robinson MD 1821-1875 Proprietor of Bensham Asylum
Robinson, G. 1859 On the prevention and treatment of mental disorders Longman, Brown, Green, Longmans, & Roberts. 228 pages. Published 1859
"in studying the nature, and endeavouring to check the progress of, mental disease, the physician must look beyond the precincts of the asylum... No one would expect to discover the causes of typhus in a fever hospital... the most useful field of study is that constituted by the circumstances... surrounding the masses from which their victims are taken. And in like manner must the science of mental pathology be pursued in its application to preventive medicine."
This website has some information on Benshamhttp://www.mdx.ac.uk/www/study/4_13_TA.htm#Bensham
As does Tyne and Wear Archiveshttp://www.thenortheast.com/archives/index.html
They might know whether any patients records survive and if so with which record office - Northumberland? Durham?