I've tried restoring it by holding the Home button, then plugging it back in.
That does not "restore" the iPod touch.
And here. Unable to Update or Restore
I have already read and tried both articles. Apparently I wasn't using proper terminology. I was attempting to put it in recovery mode and restore from there as per http://support.apple.com/kb/HT1808 (am I using the right words now?), but it does nothing. It won't display anything other than the red battery screen. It never changes to the "Connect to itunes" screen. It doesn't show up in itunes.
So I'm supposed to leave it alone for a couple days pray to the technology gods that it mysteriously starts working again, or reread articles I've already tried multiple times to no avail. Nothing new I can try?
I read and tried solutions in http://support.apple.com/kb/TS2771 under Power/Battery, tried to put it into recovery mode as per http://support.apple.com/kb/HT1808 but it never displays a "Connect to iTunes" screen, and read http://support.apple.com/kb/HT1414 but found it unhelpful since iTunes doesn't detect my device so I can't do anything suggested in the article.
Is there anything else I can try?