You should file a claim with your homeowners insurance, and tell them that your neighbor is at fault here. Provide a statement from the fire department saying that the fire was in the other unit. Also write your own statement that you told them about the fire hazard, that they declined to deal with it for years, and that their negligence lead to damage to your primary home. That they knew this was going to happen and didn't do anything about it.

I know you said they didn't pay for their insurance, but that doesn't absolve them of their liability. You'll get your repairs taken care of, and the insurance company will probably go after them for the repair cost because you are not at fault. The emergency call wasn't even for your unit. Normally your insurance would go after the other person's insurance for damages, but if they didn't pay for insurance, well... they're liable for the damages themselves.