Are you aware that Kings Park was a psychiatric hospital? In those days it's likely that the family was embarrassed and that's why they said she died. It's also likely that she spent the rest of her life there.
I don't know if it's possible to get those records, but I do know that records from Kings Park were transferred to Pilgrim State Hospital in Brentwood, NY.
Have you tried the 1920-1930-1940 census for her? It would possibly give you a ballpark period for her death.