The broken looking component is a resistor, which is used to limit the current through the LED. Unfortunately it's value would have been encoded using coloured bands on the outside which have been burnt off.
More importantly you are completely missing a high power resistor from the middle of the board (the empty blackened area), and without that that charger simply won't work at all. If you replace the big resistor with the correct value, the charger will work, but the LED won't. The LED is connected across the big resistor, so it only lights when charging current flows, so if you replace the big resistor and measure the voltage across it, we can work out a sensible value for the smaller resistor.