As long as everything else is functioning correctly, it could be the power switch. If the power switch does not move all the way to the power on position then this can happen. It would be a good idea to clean it or replace it.
I know it’s been a bit since you asked this question, but now Ifixit does actually sell the part you’re looking for. however, since you are trying to fix your iPod for sentimental purposes, and likely want the things stored on it, I regret to inform you the memory is stored on the NAND memory IC's on the logic board. Meaning if you did replace the board, all of the iPod’s storage would be gone.