I finally heard back from the insurance company...and I won my appeal! Hooray! The final $12,000 of bills should be paid by them this month. What a relief. Their letter said they reviewed my case and found an "inconsistency in payment history." I don't really care what they call it, I'm just glad they are paying it.
My submission to them included a 6 page letter that outlined my struggles with headaches over the past 14 years, how I came to choosing a nerve stimulator, and how it has changed my life. I also included an outline of the documents I had received from them and how they were incosistent. Altogether I submitted about 20 pages of documentation. I guess that did it. I was told that appeals received from patients (as opposed to medical professionals) are taken more seriously. That's something to keep in mind if/when you ever deal with an appeal of this sort.