Of course you can install your laptop's HDD internally in your desktop PC. If only to salvage the files (which I assume you would copy to another internal or external drive) you could place the laptop's HDD anywhere in the desktop - even loose or held down with some tape or some such. If you have an available bay, so much the better. All that's important is that the laptop's SATA data/power connectors are properly connected in the PC.
But you've said the laptop's HDD is "dead". You don't really mean that, do you? If it's "dead", kaput, defective accessing any data from the disk will be very difficult if next to impossible. But you really didn't mean "dead', did you?