Is it permissible to convert donation funds, which were not spent on a deceased mother, into gold and invest them, allocating the profits as ongoing charity for her?

If the money was gifted to that woman, then it is an inheritance for her heirs, to be distributed among them according to their legal shares. You are not permitted to keep it for yourself to invest it or to use it for a charitable project on behalf of the deceased. However, if the donors specified it solely for medical treatment, then it should not be spent on anything other than what they specified, except with their permission.