iPod will not turn on even when connected to charger or computer

charlie61561, the first thing I would do is to replace the battery for sure. It is definitely at the end of its life span. While you do that also check all the connections and cables for corrosion and debris. Clean it well. The dock connector needs to be cleaned of all debris, since it is providing the actual life connection for your nano. Use a wooden toothpick and/ or some compressed air if you have it. Check for bend contacts and corrosion as well. After that, re-assemble and evaluate for further errors if any.

iPod Nano 1st Generation Battery Replacement

iPod nano (1st Gen) Battery