If the hard disk still works, you can try using it in another computer and seeing if you can pull the files off.
Can you describe the sound the drive was making? Was it like a *tick* that kept repeating every second or so? If so, that is probably HDI (Head to Disk Interference) which will damager the plater and cause you to lose data.
Anyways, if that's the case, but you can still get the hard drive to work long enough, just try as I said before and plug it into another computer and try pulling all the files off before the disk fails completely. (At that point you would have to seek a professional to restore the hard disk, but I would imagine that it's pretty costly)
If the circuit board on the disk has been damaged, you may be able to get an IDENTICAL hard drive and switch the circuit board. You may need a "star" screwdriver. I learned this trick years ago with 40MB-80MB MFM 5-1/4" drives, and I just used it yesterday on my 320GB 3-1/2" SATA drive; worked perfectly.
Now, if the circuit board smoked because the motor or head-positioner was shorted or otherwise bad, you'll probably kill the new board too. Without seeing the drive and where the burn is, it is hard to be certain.
How important is the data on the HD in question? Is it mission-critical work or a collection of stuff you've downloaded over the years. Anything downloadable can be replaced. Mission-critical work *HAS* to be recovered.
First: Do not use the external HD box again. Throw it away. Now. The power supply may be faulty. External boxes are cheap enough to be disposable in this day and age.
Secondly, presuming your data is 'mission-critical': source a replacement controller board for your HD if you feel confident working with electronics *OR* take it to a professional data-recovery service now. As jtt283 pointed out: "if the circuit board smoked because the motor or head-positioner was shorted or otherwise bad, you'll probably kill the new board too." In which case your only option will be to swap the platters out into an identical replacement drive. This kind of work *HAS* to be done in a dust-free environment and even then you're lucky if the drive lasts long enough to get your data off it.
I'm sorry that's there's no simple answer to your problem. Your data can be recovered. But *please* do not attempt to open up the HD yourself. Leave that to professionals who do data recovery work.
Hiyas prtka. There should be a label on your HD which identifies the manufacturer. You should contact them first about buying a replacement controller board for your HD. If they are unable to source one for you then your next option will be to buy and identitcal make and model of HD and swap the controller boards.