A friend of mine has just received a death certificate for one of his ancestors, the date and place of death are given as: "Seventeenth of September 1844 in the Asylum attached to the Workhouse Sheffield".
I am presuming that at this time the Sheffield Workhouse would have been located on Kelham Street, a former Mill building.
Wadsley Park Village Online (Link) has an extract from:
MIDDLEWOOD HOSPITAL SHEFFIELD A Brief History - 1872-1995 Souvenir Edition - "The Sheffield Uni