Is the amount owed to the hospital considered a debt on my deceased mother, knowing that she was in a coma upon admission? And is it permissible to repay this debt from private funds later by placing it in the mosque's donation box?

This remaining money for the hospital becomes your responsibility, as you were the direct party to the contract. If the mother left behind money, you can claim it from the inheritance, provided you were not donating it. If she did not leave money and you are unable to pay it, you remain liable for it unless the hospital absolves you. As for paying it to the mosque fund, it is not permissible because its owner is present and is claiming it from you.