Well, the iPod is truly dead. This is what I see when I plug it into a power cord:
Very sad, I know.
So, I found out today (after already ordering a new one) that you can take your dead iPod to an Apple Store for a 10% discount on a new iPod. I wonder if they will give me a refund if I've already purchased the new iPod through the online store. Worth an ask, don't you think?