You can take out the hard-drive from the desktop, install it into an external enclosure like this (make sure you get one for 3.5" drives), or borrow another monitor to copy the files to a usb drive from the desktop, or setup file sharing on the network.
you can remove the hard drive off your laptop and put it into an enclosure.Plug it into your new PC as a slave.Then you can copy your files off it.
If it shows up in 'my computer' as a drive letter(like E: drive) but you can't access it,you can recover your data with data recovery program.Tenorshare data recovery is a professional one that can recover data after deeply scanning your hard drive to look for lost data due to drives crash, system sabotage, virus attack,etc.You can get it here: http://www.any-data-recovery.com/product/datarecoverypr...
After you recover your data,you can format it to check out whether it can be fixed.
Can get it recognized?
Turn the computer off without the external hard drive being plugged in.
plug the external hard drive in a usb port, power up, the systems post boot up will find the drive and inform windows what hardware is installed.
When windows boots up, it will look at all the hardware and find the external hard drive, when windows finishes booting up you should be able to see the drive.