CJ, I didn't have any luck searching for a Rose, Carolina, Ros*, Carol*, Hussong, H*ng, with and without a last name and various birth years, no birth year, died in Islip, in Ancestry's 1880-1956 Death Index. I rather doubt Rose would have lived beyond 1956 living in an institution.
Is there a chance a child or relative removed her from the hospital?
Have you looked at Albert's 1912 marriage record? I wonder how his marital status was recorded.
Police and/or court records would have been generated when Rose was committed. They won't help with her death information, however.
There was an entry in the death index for a Caroline with no last name who died in Islip in 1949, but I doubt that is Rose.
I suppose you have surveyed your aunts and uncles for any information they might have about Rose's demise.
Assuming she died at the hospital, I wonder if she were buried on the grounds or if her family arranged to bury her elsewhere. I wonder if there was a funeral for her and If relatives attended.
It appears the hospital neglected to file a death certificate for her.
I wish I could be more helpful.