Will I have paid off my debt by giving the full insurance amount I received to the dorm supervisor, instead of part of her due amount which she did not receive from the landlord, because I was embarrassed to tell him about the damage to my bed which he did not notice upon delivery?

The tenant is liable to guarantee what is damaged due to his transgression or negligence, but not due to normal use. If the damage is due to normal use, the tenant is not liable for anything. However, if it is due to transgression or negligence, he is liable for the guarantee. In this case, he must seek absolution from the landlord or return the money to him, even without his knowledge. It is not permissible to give it to someone who demands money from him, such as the supervisor, because he did not authorize that, and the tenant is not authorized to do so by Islamic law. If he gives it to her, his liability is not absolved except by seeking absolution or compensation. He has the right to reclaim it from her and demand what he gave her.