I have another TEST - that you MAY be able to do IF you have a USB-based power adapter and USB cord - such as what you might have for some Tablets or cell phones.
That TEST is to connect the USB cord to the CRADLE (not to the power adapter yet), and verify that the device STAYS ON.
Then remove the device, connect the USB cord to the power adapter (plugged in), and then put the device back in the cradle.
If the device STAYS on, and even takes a charge, we will KNOW that the original power cable is the issue.
BUT, of the device still turns OFF, then we will KNOW that it is the cradle itself that is the issue.
Does that make sense?
If so, do you have such a USB-based power adapter available to you - to do this TEST?
If you have to disconnect from our conversation - just come back when you can - so we can continue.