Is it permissible to take the difference in the price of gold from the borrower, knowing that the loan was a sum of money and not gold, and that the wife sold her gold to lend to someone, then the price of gold increased?

The implication of the question is that she loaned him the price of the gold or part of it after she sold her gold. Therefore, the rise or fall in the price of gold is irrelevant, because the debt is in currency and not in gold. A debt is repaid with its like. The Islamic Fiqh Academy has decided that what matters in repaying fixed debts in a certain currency is the like, not the value, because debts are repaid with their likes.