الجواب وبالله التوفيق
The gold in your wife’s possession, if it is her property, then it is mandatory to discharge the due zakat on that gold regardless of how much debt you are under. The wife can pay it herself (by her own money or by the money received as a result of selling part of the gold) or the husband should pay it. If the gold is not the wife’s but the husband’s property, and the husband has given it to the wife just for use, then he should subtract the amount of his debt from the total value of that gold, then calculate if his remaining wealth fulfills the requirements of nisab and then pay the zakat accordingly. If after subtracting the amount of his debt from the total value of the gold his remaining wealth does not fulfill the requirement of nisab, then zakat will not be wajib upon the remaining meager amount of wealth. This ruling is regarding debts which do not span over a long period of time.
If the debt spans over a longer period of time then instead of subtracting the total amount of debt from the total value of the gold, only the amount that was paid over the period of a year will be subtracted from the total value of the gold. Zakat will be wajib on the remaining wealth as long as this remainder fulfills the requirements of being the nisab.
واللہ اعلم بالصواب