It is now a branch of Creighton University from Lincoln, NE. However, it is possible that that connection did not exist earlier, since consolidation of schools is common practise now. I have a sister-in-law who graduated from there in the late 70's, and I have many family living in Hastings now (I live in Lincoln).
The Hastings Sanitarium you mention MAY be a mental institution currently in Hastings. I don't know it's name, but it is an institution for mental patients and criminally insane (I'm not sure of the correct terms; hope I'm not being politically impolite here).
I think the Mary Lanning Hospital/Nursing School was created in the 1910's so this may not be the place you want to look. Hastings has an active Geneological Society and you could try to contact them. Try going thru the USGenWeb site to contact them.