The fact that the external monitor works means that the video circuitry is working.
So the faults are possibly a) the connector between the video circuit and the screen (or) b) the screen or its components.
Either way you will have to dismantle the laptop, probably removing the top plate and the screen. This is usually quite difficult. You must then work out which parts need replacing and find the replacement parts.
This is best left to a professional repair guy -- and will cost quite a lot of money.