The orphanage is discussed on page 10 of the 1982 book titled, "Adoption Agencies, Orphanages and Maternity Homes--An Historical Directory." Librarians have access to the WorldCat database, which contains the names of 147 libraries that possess a copy of the book.
There is at least one address plus the name of the order of nuns that operated the orphanage and where the order had its motherhouse.
Once in a while, I hear of an orphanage’s records being obtained by a local library or state library or historical society.
Sometimes people create web pages for a specific orphanage. In your case, it may be difficult to find such a page using Google because there are so many institutions using the name St. Joseph’s. As you may know, St. Joseph was the foster father of Jesus.