I used to be able to stream the Sleepwell Channel videos through Amazon prime video no problem on my fire tv stick but then they suddenly stopped working but other Amazon Prime videos will work. I restored my fire stick to factory default and it worked for one night and now they won't. When I click play (either resume or start from the beginning) it goes to a black screen and then back to the main screen for the video. I've restarted the device numerous times and they just won't play. Yet these videos will play just fine through the Amazon Prime Video app on my phone.
Quite possibly they are no longer compatible with your device. A first gen Fire Stick is now fairly obsolete and many things will not work on it. Make sure there are no software updates you can install but it may be time to upgrade and new ones are dirt cheap and much better.
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