I have captured and plotted some energy usage data for a DK running with the LED turned off and loop() consisting solely of:
* All we do here is go to sleep
The first chart shows the current consumption and total charge used over approx 200mS. I think you can click on the image for the original, which is much larger. You can clearly see the BLE radio doing stuff @ 30mS intervals, and there is a second set of activity at a 100mS cadence.
[NOTE: edited, new image uploaded - y2 axes scale was wrong, calculations & numbers in the commentary are correct]
The data suggests that the DK asleep like that consumes approximately 2.8 milliCoulombs per second, or about 10 Coulombs per hour. AA alkaline calls offer approx 7200-9000 Coulombs total capacity, which equates well to your expected life of ~1month on AA Alkaline cells.
Here is a detail of the 30mS BLE-driven events:
Note the change in horizontal scale (natch - it’s a detail.)