Yes there was a Memorial Hospital in Mt Vernon, Knox Co., Ohio,this building is not a rest home.
If you were adopted you may not find a record of your birth, but if your mother was born in Knox Co., Ohio you may be able to find her birth record at the Health Department in Mt. Vernon, Ohio. The health Dept. has records of births and deaths in Knox Co., Ohio after 1908. The phone number at the health dept is 740-392-2200 Good luck