Real Estate Law
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You need to keep an itemized list of your damaged property and then submit it to them with estimated depreciated value... that means current actual value, not replacement value. So if you had a shirt that was $50 when you bought it new and it was 6 months old, it might only be worth $10-15 depreciated value... So you don't get replacement value, just what something is worth right now..
Once you submit it to them with a request for payment, if they ignore you then you will have to file a small claims court lawsuit against them for property damage and have the case heard by a judge. The judge can then enter a judgment in your favor and order the landlord to pay you.
But realistically if you actually filed suit, it is very likely that the landlord or their insurer will quickly settle with you.