Well, you've asked a really great question. Actually, you've asked a few good questions here.
First, renters insurance covers property that you own under the property section and property damage that is your responsibility under the liability section. You don't own the condo, and the damage was to building property, so there is no coverage under the property section.
Under the liability section you have a different issue. My answer is based on industry standard ISO forms, which I know for a fact that State Farm does not use, however, they have equivalent language for this area of coverage. Under Section 2 Exclusions, F - Personal Liability, Number 3 property damage to property rented to you is excluded unless it was fire damage. State Farm has an equivalent exclusion. So there is no liability coverage. Bottom line, your insurance isn't paying.
Onto what her insurance paid for. She Received a settlement from her insurance company to pay for damages. She can't double dip and ask for money from you for damages paid for by them. If Kemper was denying her claim in part, they would have sent her a letter explaining which part of the claim they were disallowing. Kemper sent her a check for her to make repairs, by not making repairs, she is barred from going back and asking for more money from Kemper for failure to mitigate the damage. (It's pretty easy to say the floors got worse from 2 years of non-repair.)
Now, equally important for you is your liability outside of insurance. In the absence of a lease stating otherwise (like one making you responsible for all upkeep and repairs), you are liable for damages only if you were negligent. There is a three part test for negligence. Duty owed, duty breached, and damages caused. 1. You do owe a duty to take reasonable steps to not damage the property you occupy. 2. Did you breach that duty? I can't say, because you don't say how the leak happened. What you do say is that no one is certain what caused the leak only that there was damage. 3 Damages caused - there was damage, but we can't say that your actions / lack of action caused or worsened them. Test failed, no negligence based on the facts presented. Now, that could change if you failed to report the leak in a timely fashion or you knew or should have known the fridge was going to leak. If you were negligent and liable, by the way, it would have been for all damage, not just un-insured damage. Kemper would have asked you to reimburse them for the amount they paid. That's called subrogation.
Therefore, I'm going to say tell the landlord to take his "mold remediation bill" and shove it. If anything, I would say by claiming damages for work that, is not your responsibility, deferred by the owner which exacerbated the damage, then presenting a non-itemized bill for work done themselves (failure to show actual costs), that you may have a claim for double or treble damages if the landlord failed to return your deposit timely. Now, I've started to drift from my area of expertise, and you should speak with a lawyer if you intend on trying to recover damages.
I hope this helps.
Mark Walters, CPCU
CS Insurance Strategies
http://insuresavvy.blogspot.com