Is it permissible for my sister to bathe with her friend due to the intimacy between them, knowing that if her friend becomes ill, my sister washes her, and is there anything in that which displeases Allah?

A Muslim woman's awrah (private parts) in relation to another woman is the area between the navel and the knee. It is forbidden for a woman to look at the awrah of another woman, which makes your sister's and her friend's action forbidden. Therefore, you must advise your sister to avoid this action and explain that it is not permissible for a woman to see what is between another woman's navel and knee.