I think that the "insane" term could be applied to the doctor. Yet, I would also suggest that it might help to understand the full meaning of the story if labels like "sane" and "insane" are put aside for an understanding of how individuals, driven by a desire to appropriate the world in accordance to their own subjectivity, can act. The doctor is not insane when he begins. He is calm and rational, seeking to find the cause of why the girl is in pain. Infact, he begins as a typical doctor, understanding the need to find a cure or to understand the nature of those who are ill or sick. The doctor begins trying to understand "what is the matter." It is from here that the war of wills between both sides emerges. The doctor is so driven to succeed and to overcome the will of the child that he emerges as someone who demonstrates almost a sadistic joy of exerting power over someone else:
The worst of it was that I too had got beyond reason. I could have torn the child apart in my own fury and enjoyed it. It was a pleasure to attack her. My face was burning with it.
Williams' pointing out of the "pleasure to attack her" in the doctor's mind is where he is suggesting that there is a dark side to what the capacity of human cruelty. This is not as much "insane" as much as it is an exertion of power, an ability to appropriate the will of another with the use of force as a part of this. When the girl lunges towards him and attacks him in the end, it is a response to the fact that the doctor has reached a level of depravity that is not necessarily something only he demonstrates, but something that could lie at the core of all human beings and revealed under the most visceral of conditions. It is here where I think that the doctor acted in a manner that could be seen as "insane," but really explored the dimensions of the cruelty that comes with power and the ability to exert it without hesitation or limitation.