If you connected the battery to the 7...12V in and the LED between +5V and ground, you can have damaged only the 5V regulator. If now you connect ONLY the battery to the 7...12V in, the Nano becomes hot? What becomes hot exactly?
A LED works at about 1.6...3.5V (it is different for different models and colors) and, over that voltage, curent raises very fast! A resistor is needed to fix the current to a right value.