I have tried restarting the computer, updating the driver for the second monitor, changing the resolution, et. al.

Picture (from my shitty cellphone): http://i.imgur.com/Hs2AuxX.jpg

The monitor is listed as a 'Generic PnP Monitor' in device properties and 'R912E' in Screen Resolution. Brand is Rosewill. Looks a lot like this one.
Any ideas?

The driver has pretty much nothing to do with it. Make sure graphics drivers are updated and verify that the cable isn't the cause. Swap the cable on monitor 1 and monitor 2 if possible.

Generic 1st round advice: unplug everything, take out the graphics card, clean off/blow out any dust, reassemble.

I just swapped it out for a landscape monitor and it worked fine. Looks to me like that's a pretty standard issue with that monitor upon perusing the internet, so I'm not going to use it.