First off the tech did not make that diagnosis based on the age of the battery and no recognition from iTunes only.
Then WHAT did the tech base the diagnosis of a "dead battery"? What tests did the tech have you do on a phone call? You told us (reluctantly for some reason) what actual symptoms you had, after I asked a few times. What ELSE was there...
Second it was connected to power and there was a blank screen in all situations that I tested.
Yes, you said the previously... And I'm saying that no ACTUAL tech support person with half a brain would diagnose a "dead battery" based on THAT symptom. The most likely possibility is a bad dock connector (on iPod), after first ruling out the possibility of a bad USB docking cable and a bad USB port on the computer; rule out bad USB docking cable by trying another docking cable (or using the current docking cable with another iPod), and a bad USB port by trying a different USB port and confirming it works with another iPod or something similar like an external drive or flash drive.
Other likely possibilies are bad logic board and water damage. ANY of those possible causes (and more) could result in the iPod being dead ("blank screen") when disconnected or connected to power, AND not have iTunes recognize it. And these possibilities are FAR MORE likely than a "dead battery" because your described symptoms do NOT fit simply having a "dead battery." If power is not reaching the battery, it will not get charged whether the battery is good or bad. If the iPod's dock connector is bad, the connection is never getting passed that point.
But hey, I'm the one who asked those extra questions (that for some reason annoyed you so much), and advised that the "dead battery" diagnosis is probably wrong, and first suggested that you should go to the Genius Bar to have the nano looked at in person. So you're welcome... 🙂