Your landlord, unfortunately, may not be legally liable, or responsible to pay. Person A--including a landlord--is not automatically responsible for person B's loss, simply because B's loss occured on A's land or even due to some of A's property. Instead, there must be fault--that is, A must have done something wrong. If the fence was badly built or maintained, rotten, falling down, leaning, etc. and the landlord knew or reasonably should have known that, they would likely suffice to make the landlord liable. But if it was a well-built and -maintained fence that simply succumbed, without warning, to a heavy storm, there is no fault, and therefore the landlord is probably not liable; he did nothing wrong.