The problem with an external drive, is there is no real way to test the housing, if the housing has developed a fault, no amount of connecting will get the drive connecting properly, the only way to test an external drive when you can't connect to it is to remove it from it's case and attach directly to the PC.
Now I fully understand your hesitation with going inside the case, and there is a cheap(ish) way to avoid doing that if your happy to spend a bit of money, it is simply a Drive Adapter, you connect that to the PC via USB then connect a drive to that, pretty much does the same thing, but without going inside the PC, the downside is there is no guarantee it will work, because we have yet to find where the problem is, so you could be spending money to end up exactly where you are now, if your constantly helping mates then it's not a bad tool to have, but for a single use you have to ask yourself if your happy to take the chance.
That is the cheapest way to go for a 1 off, there are other more purpose built dock's, which are great tools, but if you are only doing it the once then they are a bit expensive to get, which is why I suggest this cheaper option if you wanted to go this route, at least if it doesn't work it wont hurt the pocket as much, but do take note, make sure it comes with all you need, namely the connector for your drive, e.g. Sata or ide, but most of these solutions probably have both types included, which is the better way to go, you never know what the future may throw at you, and make sure it can power the drive as well, because the drive obviously needs to power up to work.
Should you decide to go inside the case, then I suggest an initial look without touching anything, we need to know what type of hard drive the "problem" drive is, Sata or ide, then we need to know what connections are available inside your system, and more importantly in a way what cables are available, if your "problem" drive is ide, and your board etc. all Sata, you may need to get extra cables anyway, so we do need to know what you have to work with if going inside the case, if you want to check this out.
Power down the PC, turn off and remove the power lead from the wall, then remove the side, you may find it easier to manage laid on it's side, then take a picture and upload it maybe, hopefully the picture will catch all the connections and bays, and if you can take note of what the motherboard is (make\model) and let us know that as well, it's nearly always written on the board, and take note to include if it shows as a specific version, I'm just trying to cover all the bases so you don't have to mess about too much.
Of course if you decide to try the adaptor 1st then you don't need to do all this, but if you do look inside for the info then once you have that info, you just replace the side, plug it back in and power up again like normal, just let us know how you would want to proceed we will do what we can for you, if you are unsure of what adaptor to get (assuming you opt for this), then let us know what hard drive it is (make\model) and link to what you think would do, we can double check it has all you need if it helps, hope this helps.