Thank you all for your advice. Regarding the fence issue, the reason I am so resistant to build a pen is because my property is already fully fenced, I have a Fort Knox chicken coop, and I have dogs that take care of all predators but other dogs (they're Golden Retrievers...they've never met a strange dog or person they didn't like
)
Anyways, I talked with the mayor last night, and he said that he has been having a lot of problems with this family as well. The sheriff had said the same thing. While the sheriff can't legally do anything, the mayor said that if the dog trespasses on someone's property that is outside city limits, it's fair game. The mayor said he had several friends that had seen the dog before and he said "don't worry about it anymore." I haven't seen the dog today, so it sounds like I talked to the right person.
I also talked with the landlord of the neighbor, who I am friends with and who has rescued a chicken of mine from their dog before. He agreed that the situation needed to end and told the neighbor that she was no longer allowed to keep dogs at her house - if she does, she will be evicted.
So while it's not exactly the outcome I was hoping for - I would have liked the neighbor to have taken responsibility, one way or another, and paid for my chickens - at least the dog shouldn't be coming around anymore.
PS, amenfarm, an LGD is a Livestock Guardian Dog, a group of breeds that were bred to protect livestock.