Time machine doesn't see external hard drive on network

Question

I have a pair of external hard drives connected to my iMac, which is connected to my 2WIRE modem/router via Ethernet. I'm using one of them as a Time Machine drive. For the past couple years, I've been backing up the laptops in the house to that Time Machine drive without fail.
Now none of the laptops (a MacBook and an iBook) won't see either external drive connected to the iMac, even though the iMac has file sharing turned on (as it always has) and both laptops are connected to the iMac via the Finder. Both laptops can see the iMac and the external drives connected to it via the Finder, but in Time Machine, when I try to select my Time Machine drive, nothing appears in the window.
I was having a problem with the Time Machine drive and had to wipe and reformat it, but I don't understand why Time Machine won't see the other external drive either, even if there is a problem with the TM drive. I recall Time Machine picking up all available hard drives as possible backup destinations, as long as they're also seen in the Finder.
The iMac has been having some OS X issues. The Finder will crash sometimes, and I have no choice but to hold down the power button to shut it down and then start it up. When it starts, it takes longer than it normally does, I assume because it's checking and/or fixing some things.  However, those issues have been going on for a long time (I think I need to reinstall OS X in it, since Disk Utility finds nothing wrong; also, the hard drive in the iMac crashed about 2 months ago and I put a new one in; I then restored the TM backup to it, and the OS X issues have persisted, which tells me it's software-related, not the drive).
Also, the iMac backs up to the Time Machine drive just fine. Of course, the TM drive is connected to it locally, whereas the laptops have to get to that drive via wireless file sharing.
I did try plugging one of the external hard drives on the iMac into my MacBook, and Time Machine saw it immediately, so my guess is the issue must lie with my router and/or the iMac, but I don't know what to check next.  I've restarted the iMac, I've restarted the laptops, I've turned TM on and off on the laptops, and I've turned file sharing on and off on the iMac; none of those things reestablished the connection.
I suppose I could just go with external hard drives for the laptops, but my wife moves her iBook all over the house, so I know hers won't get backed up regularly. That was the nice thing about the system I had in place, which I realize wasn't as elegant as a Time Capsule.
Any advice is appreciated.  Thanks.

Answer 1

That should say "Now none of the laptops (a MacBook and an iBook) will see either external drive..."  I hate typos.
Anyway, after rereading my post, I'm starting to think I just need to reinstall OS X on the iMac, given the other issues it's experiencing.  Any advice for doing that would be appreciated, if that's indeed the solution. Thanks.