Strange failure of wireless keyboard

Question

The wireless keyboard on a Mac I support has failed, strangely.  Some keys work correctly. The single-quote key does nothing. But clicking the "g" key will result in a single-quote appearing -- usually many of them, repeating endlessly --or as long as I could wait-- until I press anither key.  Sometimes the single-quote appears singly while typing a string completely unrelated characters.
This an aluminimum keyboard, product ID 0x239, firmware revision 0x50, supplied with a mid-2010 iMac Core i3 3.06GHz (imac11,2).   No physical damage seems to have preceded this problem.  The keyboard appears to be connectiong normally. Tried putting in new batteries.  Checked Universal Access for keyboard: everything is off.  The wireless Apple trackpad used with this iMac is, meanwhile, functioning normally.
Choosing "Setup Bluetooth Keyboard" at the bottom of the Keyboard Control Panel leads to an endless unsuccessful search, which seems strange, but maybe this fails because this keyboard is already connected (linked?  paired?) -- or because the keyboard is crazed and not responding consistently.
What's the best way of proceeding?   An old wired keyboard I connected to the system works fine.
But the owner prefers the smaller, wireless type.   I have a spare wireless keyboard that came with a Core i5 iMac that's in  operation on a nearby desk.  If I understand correctly, that keyboard was delivered pre-paired with the Core i5 machine.  Is it worthwhile de-pairing it and then attempting to pair it with the Core i3 machine?     If that combination works correctly, then I'd guess it is time to declare the failing keyboard hopelessly demented and put it in the recycling bin.  Yes?   Or, what?
I'm a hardware/software tekkie, but I don't know anything about Bluetooth or these keyboards.
TIA

Answer 1

To add to this discussion and maybe to help others. Over the past 6 months I have been having on and off problems with the Apple BT keyboard and number pad. Sometime it will not responde and other times all I would get are the strang charactors. Other have seen both on this keyboard as wel as the keyboard on Macbooks.
There has been a couple of course of actions that people found to resolve the issue. One is to turn off the computer and then restart, this cause the keyboard to reconnect with the computer which may or may not fix the problem. Other had luck doing a PRAM reset on boot up. Other simple just unpair and repaired the keyboard. I personally just pull out the battery for a 5 or 10 seconds and then put them back in and power up and it will repair and reconnect and the problem goes away. I personally think this is some sort of BT communication software stack issue, I have seen similar issue on cell phones with BT devices. Somehow the BT connection gets corrupted and the only way to clear it is to disconnect/remove power and repair the device.
If the above has not solve the problem then you have to look at the hardware. Some people reported that their keyboard was dirty and keys were sticking specifically the cmd and control keys which make sense since they are the lowest on the keyboard and dirt will migrate there. Since the key travel is so small it make sense to clear your keyboard often to get the dirt and dust out of it.