Amazon Video app no longer launches

Yes, if that method doesn't work, a factory reset would be my next step too. If you have an external hard drive, you can move your other games and apps to it first so you don't have to reinstall everything.

I've tried both methods, but the app still won't launch. I'm starting to think that I may have to refresh the machine to factory settings and reloading it.

While it didn't help me in my case, yours was a useful response, so I'll mark it as the answer.

In general when apps don't launch, holding down the power button on the front of the console for about 10 seconds until it shuts down, waiting a few seconds, then booting it back up can help. If not, uninstall the app, turn it off, unplug the power cord
from the back of the console for 2 minutes, turn it back on, then install it and try again.

Ah! Good point. Thanks for your help!

Some more info, adding to my above comment:

As an experiment, I turned off "automatic sign-in", then turned my console off and back on. I exited the sign-in prompt without signing in, and found that all of the squares in the "games" and "TV & movies" sections loaded. I then went to the "my video
apps" section and launched the Amazon Instant Video app, which launched fine, but asked me to sign-in to Xbox Live, which I did, and the app proceeded to work as normal. I then exited Amazon Instant Video, which took me back to the "my video apps" section,
from which I launched Netflix, and it worked properly too.

But I then went back to the main dashboard and found that now that I had signed in, the upper two squares in the right columns of the "games" and "TV & movies" sections no longer loaded (just showed spinning circles, as described in my earlier comment above)
and then I could no longer load Amazon Instant Video (console gets stuck on a black screen).

I've tried this twice now and got the same results: it seems that as long as I don't go to the "games" or "TV & movies" sections while signed in, so that I avoid trying to load the problematic squares while I'm signed in, then I can launch the video apps
from the "my video apps" section (the first app I launch will ask me to sign in, but as long as I then stay within the "my video apps" section, I can launch any app in that section without encountering the black screen problem).