I have had this same problem with my iPod Nano myself and I ended up tossing the iPod. The power button has a small tiny black dot of metal that sits between the metal button and the actual thing that registers a press of the button which tends to be dislocated after regular use.
See the tiny red circle toward the top right of the photo below.
In order to fix this you have to replace the entire volume button, power button and headphone jack assembly. This is not an easy fix by any means! The flex cable assembly is so delicate that even holding it the wrong way can tear the assembly into 2 pieces, and I learned this the hard way, twice! I eventually threw in the towel after spending over 20$ on replacement parts and pulling out some hair over frustration.
Unless you are 100% confident in yourself on fixing extremely delicate electronics, don’t do it!