How can I measure the charge of a rechargeable cell?
How can I measure the charge of a rechargeable cell?
  I'd like to know if the cells are actually charged.
  The battery specification is: AAA, 1.2V, 1000mAh.
  Once charged, my voltmeter reads 1.43V on each cell.


You cells are fully charged.
 I assume that they are NimH (Nickel metal Hydride) cells but that voltage would also indicate full charge for NiCd cells.
 I use a nominal figure of 1.45 V  / cell at 25 degrees C as end point when actually charging. If they measure 1.43V/cell when off charge that is even higher than I'd usually expect.
 If that is an on-charge figures at end of charge it indicates full charge. 

A system intended to operate from NimH cells should work down to 1.1V/cell in all cases and ideally down to 1V cell. When lightly loaded (say C/10 load or less) NimH cells will operate at about 1.2 V across the major part of their discharge cycle. The keyboard should load them to far less than C/10. 

Likely options in order of decreasing probability are


Your bluetooth keyboard is faulty 
The red light indicates some other problem such as a lack of link connection from the "dongle" (aka receiver) that it connects via.
The keyboard needs Alkaline cells
(but it should work on well below that voltage even with alkaline cells).


Thoughts:

I assume that the keyboard has a USB connected "dongle" which plugs into a USB port on your PC.   


You do not say if this worked previously.
If it was working and now isn't then a fault somewhere seems likely. 
If you have not rebooted the system try doing so. 
Try to minimise connection of other equipment which may cause interactions. 


Sometimes other equipment plugged into the same USB hub causes problems. 
Sometimes even equipment plugged into another USB port on the PC can cause problems .

On some occasions a device will "enumerate" on one USB port and then not operate on any other port.
Did the device came with drivers and did you load them?
Try it on another PC.
Is there a channel setting option which has been accidentally altered (most unlikely with Bluetooth). 
Do you have other Bluetooth or wireless based equipment in use which may be interfering?