We debated what to do for a bit and eventually decided that since everywhere else we really wanted to go was at least 10 hours away by bus, we'd basically lost the entire next day, thus making the trip out to Cappadocia not so feasible (out of time). So we insisted on a refund for our ticket, which took a good long bit of arguing with them, then booked ourselves on the latest bus to Istanbul that would arrive during daylight hours, then checked into an overpriced hotel that the bus station folk directed us to. All this ended up taking us until after 3:00, so maybe we should have waited for our original bus, but who knows of it was really going to show up at 3:30?
The 10-hour bus ride to Istanbul was not as miserable as we expected. It was a comfortable "techno" bus with little TV things at each seat like on the intercontinental flights (unfortunately everything was in Turkish), and WiFi. Also appropriately timed food/bathroom stops and snack service. Had there been power for keeping stuff charged, we could have made much more progress on catching up on the blog, but it was nice to be able to book our hostel in Istanbul and download enough podcasts to keep myself in interesting radio programs most of the trip.
The bus system in Turkey does have a very good reputation for comfortable buses (definitely were that), and running on time, so apparently we just got really unlucky.