ipod nano 6th Gen not turning on

Does the iPod itself indicate that it is being charged? There should be a lightning bolt in the display if it is.

If the lightning bolt does not appear when you plug the iPod into a power source, you need to check:

• the cable (try another one if you can. Perhaps borrow one from a friend)
• the charging port - to make sure there is no debris in there which might be stopping a proper connection being made
• that the port on the computer is working and has enough power to charge the iPod. USB ports on keyboards usually do not
• that the charger is powerful enough and is working. The specification on the charger should tell you how many mA the device can provide. It needs to be at least 500mA, or 0.5A

I suggest you start with the cable. Is the cable obviously damaged in any way? Even if it is not visibly damaged, it might still be faulty.