You should be able to get the hard drive out of the laptop. To do so, I would go to the manufacturer's website, and there should be some instructions, such as "upgrading your hard drive" or similar under support for your laptop. But to use it with your desktop, you're likely going to need an adaptor. You'll need to use a USB adaptor or an adaptor for connecting a laptop hard drive to a standard IDE or SATA connection on your motherboard. I'm guessing it will end up costing at least $20 to do that.
Another option (I've done this before) that does not require working with the hardware:
Get you a free copy of a Linux Distro that will boot from the CD-Rom, and actually run from that CD-Rom. It does run slower than hard-drive based OS, but it works. Then, you'll have to mount Windows in a certain way in order to get in and look at the files on your hard drive (b/c they are otherwise totally locked from you being able to read or write to them (you can see them anytime, but can't read from them or copy them until you mount the Windows stuff). I used Gnoppix in this way, and once I found out how to mount the Windows partition, I was in business. All you have to do then is transfer the files over your network to your desktop, and then reinstall!
If you want to do this, I would first ask around to any "techy" people you know to see if they have a copy of Gnoppix, or some other free Linux distro that will boot and run from CD-Rom. If you can't find any there, I would go online, and search around, there are multiple places to download those from if you have broadband internet (dial-up would simply take too long to be practical).
If you get a copy of Linux, and are able to get it to boot and run from there, and can't find how to mount the Windows partition, post back, and I'll find that again, and put it here for you. It was actually much easier than you would think, and Linux was way easier to setup for networking than Windows was!
Let us know what you end up going with.