It seems unusual that a person from Athens Co. would have been admitted to a hospital 200+ miles away in Lucas Co.
Perhaps it was the Lucas County Infirmary. That facility began in Jan. 1838 and is described in the book, "Memoirs of Lucas County and the City of Toledo" by Harvey Scribner. On page 109 it says, "it constantly cares for some 300 sick, infirm, insane, epileptic and idiotic people." The Infirmary was in the same general area, ca 1900, as the Insane Asylum.
There is a non-circulating book of Infirmary Registers, March 1855-March 1882. Several years ago I read portions of this book and I believe the registers list the person's name, date admitted, his/her physical condition, date released (or died) and in some instances, basic information on his/her family.