If it's the same version of vista (like home, or platinum w/e) then yes it will work, just use the key that is on the laptop now, now your home machine key.
Of if u would like to try cleaning it up first,
CCleaner is a great program for removing allot of useless files.
Make sure your anti-virus program is up to date, and do a complete scan. If you have none, AVG antivirus has a free version, which works just fine.
Also make sure you do the same, update and scan for spyware / malware. Free programs are also available for this. Spybot Search and destroy, is one of my preferences.
Also go through it yourself, uninstall any un-wanted programs, delete files that are no longer needed. Etc.
And as mentioned above, defragment the hard-drive, after the 'cleaning' in complete. To organize all the leftovers.
Good luck with the laptop
=)
-Joseph