Is it permissible for a daughter to perform Hajj on behalf of her mother who is afflicted with insanity?

It is not permissible for you to perform Hajj on behalf of your mother while she is alive and insane, because insanity is not despaired of being cured. If she dies and leaves behind enough money for Hajj, her guardian must appoint someone to perform Hajj on her behalf from her money. This takes precedence over people's debts and expiations. If she does not leave money, it is recommended for her guardian to perform Hajj on her behalf, but not obligatory. This is the view of the Shafi'is, and their evidence is the hadith: "The debt owed to Allah is more worthy of being fulfilled." As for Abu Hanifa and Malik, Hajj is not performed on behalf of a deceased person unless they made a will to that effect, in which case it would be from the one-third (of the estate). The rights of Allah Almighty take precedence over the rights of people, so one begins with Zakat, expiation, and Hajj for the deceased. If a debt owed to Allah and a debt owed to a human being converge in the estate of the deceased, the sounder view is to prioritize the debt owed to Allah. It is not valid to appoint someone to perform Hajj on behalf of a person who became insane after Hajj became obligatory upon them, because they might recover and perform Hajj themselves.