Hand Washing For A Sick Person
Does a person who is suffering from a life-threatening illness (can hardly take care of their own basic needs, for example, can’t use the bathroom without assistance) need to wash Negel Vasser at the kitchen sink (or with water brought to her from the kitchen sink), or is a rinse of the hands in the bathroom sink sufficient?
From your question, it is not clear how the washing affects the life-threatening illness.
Generally, In the morning after they wake up, they should be brought Negel Vaser from a sink outside of the bathroom in order to wash three times on each hand from a kli.
After using the bathroom, if no other choice exists, there is room to be lenient to wash without a kli in the bathroom after the toilet is closed, rubbing and drying outside.
When possible, they should be brought water from a sink outside of the bathroom.
See here for details:
Does one need to wash full Negel Vaser after using the bathroom even for “Ketanim”?
Is it okay to wash one’s hands in the bathroom after using it? Does it make a difference if the seat is covered?