This water damage is the landlord's fault for not inspecting and maintaining the property. You are not at fault. Your renters insurance does not cover the landlord's building. The landlord's insurance covers his building, but not for neglect or long time leaks. Long time leaks also called "seepage and leakage" are not covered under any insurance policy. That is because it is a maintaince issue that should have been taken care of and it wasn't.
No policy covers wear and tear or maintainance. So, if a pipe broke in the wall and caused damage then the landlord's insurance does not pay for the wear and tear on the pipes ( old pipes that break). The landlord's insurance only pays for the damage the water causes, warped floors, damaged floors, walls, ceilings, ect. Landlord's policy does not cover worn out pipes, no policy does.
Your renters liability policy only covers you for things you are negligent for. You were not negligent here.