I'm guessing the battery is fried, in which case you might look for 3rd-party battery replacements--but the installation process may not be something you want to tackle yourself. And you might find the battery isn't the problem (or isn't the only problem) and then it's wasted money/effort.
Maybe your best bet is to just let it become your home stereo, always plugged into power and speakers, and get a new iPod for cheap (beware clones) off of eBay. Used nanos might get especially cheap around September when new nanos are likely to be out.
My only other thought is, wash it again, but try a different detergent.