Just an update, guys.
Thanks for all the suggestions. We called Kobo support, and the agent recommended that we make sure the Kobo was fully recharged. She said it wouldn't show up in Windows if it was still charging. So we waited the requisite 3 hours for the light to change colors, but still no dice.
We ended up mailing the unit back to them, and they send us a new one (or fixed it and sent us back the same one - we can't tell for sure, but it works anyway).
They were incredibly fast at this. After we called and they decided we should send it in, we had a kit couriered to our doorstep in one day. We shipped it the next day, and 2 days later, it was back. Now it works again.
It sucks that it had the problem in the first place, but kudos are owed to Kobo support for their great support and incredibly fast turnaround fixing/replacing the unit.