Is it permissible to borrow from the money of a mother afflicted with Alzheimer's disease to repair a car, with the intention of repaying it later?

The matter of your mother must be referred to the Sharia court to decide on placing her under interdiction, as it is the only authority authorized to do so. Until the ruling is issued, her money must be safeguarded. If the court rules that she is competent, then her permission is required to borrow from her money. If it rules to place her under interdiction, then the guardian must not dispose of her money except in what is most beneficial for her, and it is not permissible to lend from her money or give charity from it, even if she orders it, because she is under interdiction.