Here's a citation of a directory :
UNITED STATES. Census, Bureau of the. "Benevolent Institutions, 1904." [Washington, D.C.: Govt. Print. Off.], 1905. 335 p.
Idaho didn't publish many directories that I know of. The best research, I think, would be the federal census of 1900. It's indexed by the Soundex system. 99% of the 1890 federal census was destroyed by fire, and Idaho did not conduct any state censuses.
A census would include the inmates of any institution.
As far as finding old records of orphanages, don't be too optimistic. But check a state or county historical society, in case one of them inherited the records you seek.