Is it permissible for me to demand from my sick maternal uncle the expenses of the Hajj that I gifted to him, so that it may be a valid Hajj and recorded for him, even though I know I am not in need of the money?

You are not permitted to claim Hajj expenses from your uncle because you volunteered to perform Hajj on his behalf, and your Hajj is valid as a voluntary act. It is important to note that for Hajj performed on behalf of another, the person on whose behalf Hajj is performed must be obligated [mukallaf]. If your uncle was obligated to perform Hajj before his coma, then it is permissible for you to perform Hajj on his behalf; otherwise, it is not valid, as he is not obligated. Nevertheless, you can perform Hajj for yourself and supplicate for him so that he may benefit from the reward of your Hajj. This is the view held by some jurists, that whoever performs an act of worship and dedicates its reward to a living or deceased person, that person will benefit from it.