If you are willing, (there is always a risk) you need to take apart both computera. I would consider the best option is to put the good motherboard into the computer with the good LCD.
You can remove the hard disks and batteries externally quite easily.
I would suggest taht you do each step at the same time for both computers.
You will need an earth strap and a screwdriver (the right size, magnetic is best) and remove all the screws. The keyboard comes out quite easily and you should see the
motherboard.
Note where all the connectors go and remove them. Remove the motherboard and put it into the computer with the good LCD.
Reassemble by reversing the steps you have done.
GOOD LUCK