What is the ruling of Islamic law on a person withdrawing money from his elderly uncle, who suffers from Alzheimer's, by virtue of a power of attorney, and depositing it into his own private account, then giving some of it to his father (the uncle's brother) to spend on his uncle? And what is the way to correct this if it is impermissible?

There is no objection to requesting agency and documenting it. The money for which you have become an agent is not owned by your father; rather, it remains the property of your uncle as long as he is alive. You are not permitted to dispose of it except in the most beneficial way. Spend from it on your uncle, and what remains should be preserved for him. It is important to care for and honor your uncle, as the Sunnah dictates respecting and honoring the elderly, based on the Prophet's (peace and blessings be upon him) saying: "Indeed, part of glorifying Allah is honoring an elderly Muslim."