I wouldn't worry about the battery at this point. When the iPod is connected to power, it should operate normally (as an iPod nano and linked to iTunes) whether the battery is strong or weak. If it does not, the cause is NOT the battery.
You seem to have gotten it out of diagnostic mode, so the original problem is solved. Can you restate the current problem(s)? The nano must be recognized by iTunes because you can do a Restore on it, is that correct? When you do a Restore, iTunes says it completed successfully (no error messages), is that correct?
my computer stated the Ipod was restored, it was not.
Describe what happens next that indicates it was not restored properly. If we solve THAT problem, maybe the actual problem will be solved.