Hard to say, some discs will indeed work in another machine, most won't.
At work we've a corporate license for x amount of PCs of any M$ OS from XP up. We only have an upgrade disk for Win 7 (actually, we don't, but the boss doesn't like us doing things the easy way when he may need to reinstall whilst working at home!), so if it's a borked drive I have replaced then I stick an old Dell XP recovery disc in, install XP and then Win 7. Technically we're illegal for a couple of hours whilst I do this, but it is only temporary.
Even
if it worked, your license is only for one install so unless you destroyed your laptop install, or bought a separate license, you'd be illegal too. But not temporarily for a couple of hours.
I'd recommend budgeting for a new OS, and I'd recommend the Pro over any Home version. I honestly can't recommend spending extra on Ultimate. Pro here is running at about £100 (~160 usd).
