How can the value of a stolen and sold gold piece be restituted without knowing its weight or price, and without directly confronting its owner?

Regret is the first step in repentance. Since the gold has been sold, its value is incumbent upon you. If you do not know its weight, then take precautions to safeguard the owner's right and pay her what you are certain will clear your conscience. You are not required to inform her about the theft. Giving it away as charity is not valid as long as it is possible to return it to her, unless she forgives her right. You have done well in your eagerness to free yourself from this injustice, for the Prophet, peace and blessings be upon him, said: "Whoever has wronged his brother concerning his honor or anything else, let him seek his forgiveness today before there will be no dinar or dirham."