Hi, Check your laptop with some other adapter. If it works, then you have to check your adapter using multimeter for the exact output voltage which has marked in the adapter itself. If that is not working, kindly check the battery condition because as you are using adapter and laptop connected together always may cause your battery malfunctioning and the power circuit in your laptop. Still you have facing the same problem, open the adapter and check.
Hi, check the exposed wire and if the exposed wires are shorted then isolate with a proper insulation. Try it again for LED glow. If charging is ok then go for bisecting the adapter and check the soldering inbetween the LED, resistor and the power connection. May be this will help you.