Is my wife considered sinful or earning ill-gotten gains, given that her unpaid leave salary was not deducted despite my repeated follow-ups with the relevant authorities?

You did well by informing the responsible authority to rectify the error and deduct the salary for the month your wife was absent. It is obligatory to inform the responsible authority and return the salary if an employee falls short in their work and does not rectify it, as the wife is not entitled to take what is not rightfully hers due to the employee's error and negligence. She must strive to return the salary to the responsible authority.