Torrent bound

Less than 100 days out of the box, my iMac has lost the connection to both the keyboard and the mouse.
As of this afternoon, both devices have brand new batteries. When I turn them on, the light stays steady for a few seconds, then blinks, then goes off.
Bluetooth is turned ON.
'Discoverable' is checked.
I removed both devices from the list, thinking maybe they needed to be reinstalled. I am able to locate them, but keep getting messages like these:
'The pairing attempt was unsuccessful. This configuration of Mac OS X and Bluetooth software could be invalid or unsupported.'
'The pairing attempt was unsuccessful. Make sure your device is in range of this computer, turned on, and discoverable.
On a couple of attempts with the keyboard, I've gotten a message providing an 8-digit passcode which I'm supposed to input, but typing it on the keyboard doesn't do anything. I've tried adjusting that setting to not require a passcode, but that makes no difference.
(Funny, when I Craigslisted my old computer I had a hunch I'd better hang onto the mouse and keyboard; good thing I did, or I'd be completely out of commission at the moment.)