If you have a milliamperometer, you can build yourself a check strip by sandwiching a strip of paper between two aluminum foil strips. Place the strip between one of the batteries and the holding spring, so that one aluminum strip touches the battery, the other touches the spring, and the paper prevents the strips from touching, so that the circuit is open (and the keyboard doesn't work).
Bite the two aluminum strips with the multimeter pincers, so that you can read the actual current flowing through the keyboard. The keyboard should now work (a reading of zero current means that the two strips are touching, shorting out the multimeter. No harm done, but the keyboard can't be tested - replace the strips).
The NOA-1100 is rated for 3 mA. A Duracell Long Life AAA is rated 1150 mAh. 1150/3 gives 383 hours, that is, sixteen days of continuous operation. Simple good sense says that the keyboard should go in power-saving mode when it's not in use. Burning away that charge in so little a time as you've noticed should lead to a perceptible overheating: if it is so, the circuit itself is damaged and the keyboard has to be replaced.
And yet you say that the "flat" batteries do work OK with the mouse, so this is not what's happening.
What could be happening is that the keyboard has a voltage comparator to advise of battery being low. The battery is considered "low" when its voltage falls below a certain threshold - all comparators work that way.
But voltage against time at constant discharge is not linear (this is not the correct battery, but the concept is the same):
So if the comparator is set at 1.3V, in this picture you'll see the 120 mA discharge activating LOW BATTERY after four hours, even if the battery is capable of continuing supplying power for three times as long. If the comparator was set at 1.2V, it would not signal LOW BATTERY until ten hours have passed.
If your keyboard has a defective, too greedy regulator, that wants the battery to always be (almost) at the top of its nominal charge, it's no wonder that after a couple of days it rejects the batteries -- nor that those same batteries happily power a less finicky device.
What you can do, apart from tinkering with the regulator (supposing it's even accessible, that is; it usually isn't), and supposing you fail in having the keyboard replaced under warranty, is try to power it with a pair of AAA rechargeable Nickel-Zinc batteries. This kind of battery has a voltage rating about 7-8% above that of an alkaline cell. Problem is, these batteries - also called Zn-Ni or NiZn batteries - are not very common, nor cheap. On the other hand, they can be recharged, so in the long run there's an advantage there. Watch out for the first days to see if the keyboard overheats (it absolutely shouldn't).