Got impatient (as usual) so I sideloaded KitKat.
The tablet needs to have USB debugging turned on, and you need to have adb installed on the machine doing the sideloading (adb is in the Android SDK).
You'll also need an OTA image for the tablet. I got an official one from https://developers.google.com/android/nexus/images

Once you have the above sorted out, here's how to sideload Android:

1. Power off the tablet, and power back on by holding the Volume Up and Volume Down buttons while it boots.
2. The fastboot menu should appear and you should see “Start” in the top right. Press Volume Up till it changes to "Recovery". Press the Power button to select it.
   You will see an Android with a red exclamation point icon, press Volume Up and the Power button at the same time.
3. You will get a new menu and you can navigate to "apply update from adb".
4. Connect tablet to computer with your USB cable.
5. The tablet will display something like "Now send the package you want to apply to the device ..."
6. Run 'adb sideload $path-to-zip-file-containing android' from the computer.
7. Tablet will display percent progress, followed by descriptions of tasks it's running, like "Unpacking new recovery" and "Patching remaining system files"
8. The final line will say "Install from ADB complete."
9. Unplug USB cable and notice the menu at the top of the screen on the tablet says "reboot system now". Press the power button to select.
10. Tablet should reboot in to new Android.