Touch screen and computer

I think that is all you can do, what about just putting the laptop to sleep instead of off, screen should then be able to wake computer for use, Rhys.

Auto power up and down is the biggest problem with a laptop.

If your PC is out of warranty, and you are game the best way to do it is to take the laptop apart and solder wires to the power switch. Then you can run them out to a momentary push button switch that you put somewhere convenient in the car. You could take it to a pc repair shop and they could do it in a blink.

Having gone that far there are a few automatic power up / power down circuits.

like this

You can even go further and get one of these.

Cart PC power supply

While primarily designed for Mini ITX format pc's you can use the reset circuit for laptops if you do the above mod. That way it can be configured to automatically start, I use it to start the PC 5 seconds after vehicle start, then power down say 20 mins after switch off so it doesn't power down at a servo/quick stop etc. Also if you turn the car off while it is booting it will keep the power on long enough for a full boot and shut down.

This is fully automatic and has been very reliable.