Must I pay compensation for the metal part of the chair that I broke at school, considering that I no longer remember the exact condition of the chair, and a long time has passed since the incident?

It appears that the manager is authorized by the owner to grant forgiveness. So, if the manager forgives you and overlooks what happened and does not request compensation, then nothing is binding upon you, nor is it necessary to refer back to the owner. This is if you misused the chair. However, if the damage was due to normal use, then you are not liable.