How can I pay the zakat on my wife's gold jewelry given my financial hardship and debts?

The husband is not obliged to pay Zakat (definition) on jewelry on behalf of his wife, because Zakat is obligatory upon the owner of the wealth, and the owner of the jewelry is the wife; therefore, she must pay its Zakat. It is permissible for the husband or anyone else to pay it with the wife's permission, and he will be rewarded for that. There is no objection for the wife to pay the Zakat of her jewelry or other assets to her indebted or poor husband, as this falls under the general meaning of the verse regarding charities, and she will have two rewards for doing so: the reward of kinship and the reward of charity, as mentioned in the hadith (definition) of Zaynab, the wife of Abdullah.