What is the ruling in Islamic law regarding a daughter washing her mother and looking at her private parts, especially if the mother is incapacitated and there is no one to care for her except her daughter?

It is not permissible to look at the mother's awrah except in cases of extreme necessity. Nursing an ill mother does not permit looking at her awrah if it is possible to clean her by touching the awrah from behind a barrier, because necessity is determined by its extent and should not be overextended.