I am not sure where the old staff records are held, if still available.
After the formation of the state it was ran by the County Council and then it was under the Western Health Board from c1970 when that was formed
LINK to one statistical report.
The hospital closed a few years ago and is still owned by the HSE as far as I am aware
The HSE may be able to help as to where historic files are.
You could also try:
Galway Family History Society West Ltd.
St. Joseph’s Community Centre, Ashe Road, Shantalla, Galway City
Patrick BYRNE was already working in the Hospital when he married Norah FALLON in 1914
LINKLINK to 1911 census for Asylum showing staff including Patrick BYRNE & Norah FALLON
ARTICLE on St. Brigid's Ballinasloe's history including a photograph of staff in 1935
Photograph of BuildingLINK - There are further photographs at the National Library