If you had renters insurance, you would have some coverage for this. And that is why you always have renters insurance. But you don't.
If the landlord is negligent, just saying, 'because you don't have renters insurance, we aren't responsible' is a lie. If the landlord is negligent in some way, the landlord is responsible. You can sue the landlord for depreciated value of what you lost. You won't get replacement cost.
But you must prove that the landlord knew there was a problem and failed to do anything about it and the problem became worse to the point your property was damaged.
I don't know how you are going to do that. You can hire your own expert to come in and determine the cause of the leaks. You will have to pay for that expert. If you sue and win, you can ask for that money. You need to properly document your damaged items, photograph everything, show the damage, perhaps get other experts to state that they cannot be cleaned and are ruined. If they can be cleaned, you can ask for the cleaning to be paid for in your lawsuit.
The people that win are the people that fight hard. You can't just sit and do nothing, you must lead the fight, get experts to help you fight.