How is the number of years for which my mother did not pay zakat on her money estimated, given that she instructed me to pay it after her death, but did not specify the number of years?

Zakat is payable on the gold left by the mother for one year only, because the default principle is the exoneration of one's liability from anything preceding that, as it is not obligatory to pay for more than one year because it is doubtful.

Al-Juwayni stated in "Ghiyath al-Umam": "Anything whose obligation is unclear, the default principle is the exoneration of one's liability regarding it."

Al-Izz ibn Abd al-Salam said in "Qawa'id al-Ahkam": "Whoever doubts whether something is obligatory upon him [such as purity, prayer, zakat, etc.], nothing of that is obligatory upon him, because the default principle is the exoneration of his liability."