Tengo watched this series of movements and wondered how she must feel to live her life in this sanatorium by the sea, taking care of senile old people whose prognosis was grim.
(H. Murakami; 1Q84)
Would you be so kind to explain to me why it is not written as follows: "...taking care of senile old people whose prognoses were grim."
To your question:
This is simply a way of saying "each of whom has a prognosis which is/was grim". Putting it this way, as a literary technique, makes all the people just one mass of senility with a grim prognosis.