Is undistributed zakat money entirely unlawful, and can the mother pay zakat on behalf of the mentally ill father?

Your father is obligated to pay zakat on his wealth if he is capable of managing his affairs. It will not be validly discharged unless he pays it himself or someone else pays it with his permission. You must advise him to pay it, explaining its importance to him with gentleness and wisdom. If he does not do so, try to ascertain his wealth and the amount of zakat due on it, then seek his permission to pay it from your money or your mother's money. If he grants permission and it is paid, it will be valid. However, if he does not grant permission, it will not be validly discharged and must be paid from his estate after his death.

You must also advise him to withdraw the money from the usurious bank, even if he is not earning interest from it. It is preferable to avoid spending from this money if possible.