In reply to nickinscottishmountains:
Surely you shouldn't need to be able to start the old computer? I'd try getting a USB hard drive cradle as suggested above, physically remove your hard drive from your old desktop and put it into the cradle then plug in to your new laptop.
Hopefully, as long as the drive itself isn't kaput, your laptop will be able to see all of the files on it and it will be a simple case of drag and drop.
That's effectively what I did when my old system wasn't working. The hard drive in my case was being very temperamental so I put a second hard drive into my desktop, loaded it with Ubuntu and made that the boot drive. Once up and running I could explore my old hard drive at will and retrieve all my important files from it.