How do I get an insurance company to settle property loss claim?

The upstairs neighbor's insurer is not directly liable to you and in fact has no obligation to you: their insurer is THEIR insurer and their obligation is to defend their insured in court and/or to pay for them if they are ordered (by a court) to pay. Often--but not always--the insurer will elect to pay or offer something voluntarily, if they believe that the damaged party (e.g. you) will likely sue and win, since if they will be sued and lose, why not offer to pay and save the cost of litigation? But that's voluntary on the insurer's part: they don't have to pay claims against their insured if they don't want to, unless their insured is sued and loses in court. Therefore, the only way to attempt to force them to pay is to sue the at-fault homeowner and prove in court with evidence (by a "preponderance of the evidence," or "more likely than not") that they were in fact at fault; if you can do that, they or their insurer should pay. If you can't prove fault, however, neither they nor their insurer must pay. Fault in this context means negligence or carelessness: you must prove that the homeowner did something careless contributing to or causing the leak. If they were blameless in the leak--ie. it occured through no fault of theirs--they and their insurer are not liable (this way may be why their insurer is not paying, by the way: they may believe their insured is not at fault).
The expiration date of the policy is irrelevant: the insurer is liable for all loses caused by an at-fault insured while the policy was in effect, even if the policy expires or terminates after the loss.