Maybe you know of a library that has these books---directories that I trust will list the orphanages in Minnesota.
MINNESOTA. Secretary of State. The Legislative Manual of the State of Minnesota . . . St. Paul, [etc.], 18__-19__. __ vols. Blurb: ?Published biennially by the secretary of state, according to act of Legislature of 1889.? Includes a roster of state and county officers, state departments and institutions. (Minnesota blue book) (LC 9-34825)
UNITED STATES. Census, Bureau of the. Benevolent Institutions, 1904. [Washington, D.C.: Govt. Print. Off.], 1905. 335 p. (Special Reports). A directory of 4,207 institutions (excluding 85), with their addresses and statistics: orphanages, hospitals, temporary homes for adults and children, etc. Includes a single alphabetical index to the institutions. (Charities--U.S.) (Hospitals--U.S.) (Orphanages--U.S.) (LC 06002276)
Without specific knowledge of the disposition of orphanage records in Minnesota, I can only say that for various states I hear of records being in the custody of a state library or local historical society---on a hit and miss basis in the absence of any statutory plan for record-keeping.