Ideas for reinstalling Mac OS X on a 2009 Macbook Pro whose shift key is constantly stuck down?

I have a Macbook Pro 13 inch from 2009 on which I spilt gin a few months ago. As a result the shift key is constantly stuck down.

Initially, the problem was that I couldn't start the machine up, as it automatically went into safe mode, due to it booting with shift stuck down. I could get around this by holding the CMD button and tricking it. When it's started up, I disable the keyboard via command line and use a bluetooth keyboard from there.
Now I want to reinstall Mac OSX on it, but I can't do anything on boot; it won't boot into device selection screen or boot from CD.

Has anyone got a crafty plan on how I could go about getting OSX reinstalled on this machine? I have a 2011 Macbook Pro at my disposal also to help the case.

I was thinking doing a network install, but I still think that needs me to get the broken macbook into the device screen via holding a button right. Is it that correct?

Can you get the broken Mac to start in Target mode? You achieve that by pressing T while booting, it might work with the shift key too.

After that you can use the Mac as an external drive, so you can start the installation using the working MBP, pointing the installation to the broken Mac.