beem their before - and its your responsibility in many ways - you should have done due dilligence and checked the place over before doing anything for your animals sake - When I was renting I always went over with a fine tooth comb, had too many accidents with horns stuck in rails, barbed wire in places it shouldnt be, embedded in the floor etc - so Personally id cough up, and remember to always always check everything over.
Then again the landlord also has some responsibility, IF they KNEW the sheets were their they should of told you, that said if a previous tennent left them and they didnt know - they are not really to blame are they?
Bet to see what happens - if the tires were nearly worn out, you may find your contractor doesnt chase, or just asks for a contribution - if they were new,. well If I was the driver Id expect at least something.