My older 6th Gen iPod Nano does the same thing and has been out of warranty for some time. I will eventually be taking it apart to permanently fix it, but for now I can get it to turn on by using the USB cable or dock, which I use it with most of the time anyway.
Although the Barios solution sounds simple enough, as he emphasized, the extremely small scale of these parts and fragile construction warrant extreme caution, otherwise you will break or lose stuff. It requires very small tools, a magnifier, good light and patience. The fact that Apple chose to use shims instead of simply manufacturing the buttons in the correct thickness was obviously an oversight on their part.
I have a friend who routinely does iPod and iPhone repair as a business and he can do this quickly and relatively affordable. I actually love the 6th Generation model because of it's very small size and the ability to wear it as a watch if you want. I was saddened to learn of Apple's decision to make the new nano twice as large.