I usually use something like this:
http://cgi.ebay.co.uk/3-in-1-USB-2-0-To-SATA-I DE-Adapter-Cable-Power-SA1_W0QQitemZ2601419413 94QQihZ016QQcategoryZ74941QQrdZ1QQcmdZViewItem
(picked at random from the selection, though that is quite a good price)
You can also get non-eBay ones at a higher cost:
http://www.gizoo.co.uk/Products/PCGaming/PCGad gets/IDESATADriver.htm
Take the hard drive out of the laptop - you may also need to remove any caddy it may be in - and connect via USB to a working machine. If the older drive isn't completely shot it will show up as a removable drive that you can easily copy files from.
You should also be able to connect the 2 laptops via a crossover cable or USB transfer acble (http://www.thinkgeek.com/gadgets/electronic/8199/). Never used either method myself, though.