Did you try entering your Laptop BIOS and making sure maybe a USB legacy support is active? Also some external drives are recognized as USB devices by some BIOSes while others are recognized as HDDs, you should check if at least your external drive is recognized in either part of the BIOS, if it is, then the issue may lie with the software not the hardware, just a guess.

I have one of these from Newegg to allow me to read drives without an enclosure. Works like a champ and covers older IDE drives, too (why I got it). Yeah, it's not a permanent solution but it's good for troubleshooting and file transfers in cases like yours.

I assume that your new computer is running windows 7 / 8 with admin access. If so you can check in on any attached HD's by plugging them in then opening Disk Management.

Open Control Panel -- System and Security -- Administrative Tools -- Computer Management.‌
In the central pane, open Storage.

You should now see both the computers internal HD and the attached. If you right click on the disk name (the lower central section in grey) you can open the drives properties. You can now check what is causing the disk to be seen, but not open.

"Maybe some Windows expert can tell more, but could there be some kind of Windows user access control thingie deciding that you are not supposed to be able to access the hard drive, unless you're logged in as that user? Have you tried to access it on some other PC (using the same enclosure)? Is the HDD possibly encrypted, e.g. was it your work laptop?"
No, thats not very possible, there maybe problem accessing files due different NTFS Security IDs for file owners (easily defeatable), but there is not way to prevent actual hardware to work in another PC

"Also, in some cases the HDD might be signed for the original device, ie. it is usable only with the original device (in this case, your broken laptop). But frankly I think that is only used with e.g. gaming consoles so that you can't take the internal HDD and use it with some other console (or PC), some kind of anti-hacking technique I guess. At least the original HDD from my classic XBox is blocked from working anywhere outside that XBox, can't format it to use as a generic PC HDD etc."

Again highly not probable. There is actual ways to do things like that, but its dungerous and very rarely used.

"Oh, and is the enclosure USB-powered? In that case it is possible it is not getting enough power for the HDD to operate correctly, so you might have to use an enclosure that gets extra power (either with its own power cable, or a second USB connection). Also make sure the USB cord is connected directly to one of the USB ports, not through an USB hub or even an USB extension cord. Those sometimes make my external USB-powered HDD not to work properly."

USB usualy can successfully powerup 2.5" drives, but *NEVER* - 3.5' onec (exception is esata-p\usb hybrid ports with 12V power line, again very rare thing)

Now, summary:
There is possibilities:
1)your drive enclosure is in bad state (faulty) - pretty usual case for USB ones
2) There is hardware problem with drive itself - doubt but possible

Also, "Anyway, the hard drive (in the enclosure) shows up in Devices and Drivers as a local drive" not really means anything. I have enclosure too, its appears as drive EVEN without HD loaded, just zero size.

only real solution i can say - try to use linux (its can give more info about thats going on) or find tech advanced frind who will connect your drive to his PC via SATA link. Just because you should never, really use USB converters especially if they cheap.