My hard drive is about to flake out, and I'm trying to back everything up before it dies. However, it is now not recognizing my LaCie external hard drive. I can see the drive in Disk Utility, and I "repaired" it without it detecting any problems. Is there a way to force my computer to mount the drive?
There's a Mount button in Disk Utility. You might need to use "Window > Show Toolbar" or "Window > Customize Toolbar..." to see it if you've disabled it.
A family member I do remote tech support for had a LaCie external 500GB drive that took an obscene amount of time to mount (like she'd plug it in and go do something else for an hour and it would eventually show up). She had never reformatted it, just used it the way it came from LaCie. I had her reformat it and put a GUID Partition Table (GPT) on it using Disk Utility, and ever since then, it's been fine.
If you're running Disk Utility from a failing drive you could be seeing a "false positive" - if the drive is trying to die.
Try booting from the DVD/CD that shipped with your machine, and (if possible) put a minimal OS on the LaCie. Startup from the LaCie, with your clean-non-malfunctioning OS, and pull what you can from your dying drive.
I've had a couple mac drives die over the years. More often than not, I had to bring in an additional machine, or pull my dying drive out, and install it elsewhere as the slave.
If you can get your hands on another Mac with FireWire, you could also trying booting the failing machine into target disk mode (HOLD T while booting), and pull files from it this way.
Good luck - disk failure sucks.