For recovering the data, if you have trouble setting up a network it might be easier to just buy a USB hard drive enclosure for 2.5" drives. You'd take the drive out of your laptop, put it in the enclosure then hook it up to your desktop's USB port which would then see it as a removable USB drive. Empty enclosures are very cheap but make sure you get one for the smaller 2.5" laptop drives, not the larger 3.5" desktop ones.
As for how to see what's going on on your desktop without a monitor, that's more complicated. The laptop almost certainly doesn't have an input for video. There is software available that lets you use a laptop as a second monitor for a desktop, but a) the two machines have to be already networked and b) you won't be able to set up the software on your desktop if you can't see what you're doing! Sounds like it's time to buy/borrow a monitor.
Regards
Nelviticus