Hey there.
How did you try to change the drive's letter? Did you get any error or anything? This is done via Disk Management and not Device Manager. Here's how to do it:
How to change the drive letter assignment in Windows.
If that doesn't help and if you have access to your drive via your friends' computers, I'd recommend that you backup your most important data, just to be on the safe side. After that download the HDD manufacturer's diagnostic tool and test it for errors. Try reinstalling or updating all the motherboard associated drivers for your computer (chipset, USB controller, etc), to see if the drive is properly recognized afterwards.
You could also post a screenshot of what you see about your HDD when you open Disk...