A place of good climate for invalids

One word substitute is Sanatorium.

Asylum: the protection granted by a state to someone who has left their home country as a political refugee.
Hospital: an institution providing medical and surgical treatment and nursing care for sick or injured people.
Cemetery: a large burial ground, especially one not in a churchyard.
Sanatorium: an establishment for the medical treatment of people who are convalescing or have a chronic illness.