Jag hittade det! Korthistorien heter "Dormant Soul" av Josephine Saxton. Den publicerades i The Magazine of Fantasy och Science Fiction , februari 1969. Den här frågan innehåller också en väldigt konstig berättelse av Vance Aandahl som heter "Drool".
Berättelsen återges i boken The Power of Time . Den här länken innehåller en beskrivning av bokens innehåll, inklusive "Dormant Soul":
Från ovanstående länk:
The protagonist is a thirty-five-year-old woman who lives alone. One night she is visited by an angel, who reveals he is actually a visitor from another planet. It seems she is at risk of being possessed, or has been possessed, by demons from another planet, and Armaziel has come to free her. Part of the cure involves getting seven random people to pray for her. So she rings names she has picked from the phone book, and it seems to work.