"Those answers do not make sense, if the cause is a dead battery. If it has power, either through a USB connection or from a power adapter, the iPod should work normally whether the battery is working or not. It should work if you removed the battery. If it is not immediately working, a Reset..."
That is not correct. If you completely deplete the lithium-polymer battery of any iDevice when you connect it to power it will be completely dead, black screen, just like she has described. You need to wait 20 - 30 minutes for it to charge and then you may need to Reset it (and possibly more than once) to get it to display and boot up. If the battery cannot even take a minimal charge, i.e. it has flat-lined, then the iPod will not work even if connected to power, no matter how long you let it charge.
Yes, you could remove the battery and do some rewiring to bypass the battery and it would work when connected, but it will not work with a battery that is completely gone.
I'm sure the AppleCare representative she spoke to went through all this with her when she called. She could indeed make a Genius Bar appointment at her nearest Apple Store and they could attempt to charge it and run battery diagnostics on it assuming they can get it to take a minimal charge. If it can't they can ask the service manager about doing something for her, they can offer to do an out-of-warranty replacement, they can give her 10% off of a new one for recycling the 5th Generation.
If she was not watching videos or taking videos but primarily using it for music while walking, exercising, or just relaxing, she may prefer the 6th Generation Nano which is smaller, has a clip to attach it with out a case, and has multiple watch faces so she could wear it on a watch band.
But that is up to her.