How can the sister's share and the mother's right be determined, taking into account accumulated debts, verbal agreements, and multiple transactions between them?

The fundamental principle is that a loan is repaid with its equivalent. So, whoever borrows gold should repay gold. It is not permissible to agree beforehand to repay it with cash or silver, because that would constitute a deferred exchange, which is usury (riba).

However, at the time of repayment—without a prior agreement—there is no harm in using cash instead of gold, provided that it is at the rate of the day of repayment and that the exchange takes place in the same sitting.

As for your mother, if she borrowed cash, she is obligated to repay the same amount borrowed. If the currency depreciates by a third or more, one refers back to the value of the cash at the time of the loan, and what is equivalent to it in gold is repaid, or a reconciliation is made by distributing the loss between both parties.

The repayment must be in a different currency, or in gold or silver, to avoid usury. Therefore, one should look at what your mother has paid. Either she has fulfilled her obligation, or there remains something due from her, which she should repay in another currency or in gold.