Three AA NiMH cells typically have the capacity of over 10Wh (3 x 1.3V x 2600 mAh). At some types of alkaline AA cells it could be over 15Wh.

Only theoretically. I have tested some 2500 and 2600 mAh cells and in practice they all had less than 2000 mAh or even less.

No accus recharge alone. It is the voltage, not the charge, that recovers. During the discharge, the chemical processes between the electrodes and the electrolyte saturate the electrodes, resulting in drop of potential (voltage). It is the dropping of the voltage, not the lack of energy that causes the LED to dim.

During the rest, the battery chemically and thermally recovers, and the voltage grows back (while the charge remains the same), and the flashlight will start again with the full (or almost full) voltage and hence luminosity. You may be able repeating it many times before you completely drain the battery. Although, the cycles will be shorter and shorter.

Voltage alone means nothing, if the charge remains the same the voltage will break down immediately if you apply any load.
For a normal accu i would expect a significant brightness only for some minutes if it had been discharged some time before.

Only if there is no sufficient charge, which is not the case when you power a LED. It is different at other appliances, not so sensitive about the voltage - they can drain the accu indeed faster and close to the limit of their capacity. Not so at a LED - as I explained, due to the chemical changes, the voltage will drop before the charge drops significantly. This potential (the voltage, not the charge) will then recover again, after some rest. The voltage at chemical batteries is not in direct correlation only with the charge as you assume. It depends on the chemical state and on the temperature too.

Ok, so you run it down from 100% to 30%, LED goes off, you let it recover for some time, it remains at 30% charge but voltage increases so you can do another run which will run it down from 30% to 10%, etc... so the LED will work let's say 2 hrs for the first time, 30 min for the second time, etc...  until the voltage remains permanently below the minimum operation voltage for the LED.

I found another paper, explaining the voltage recovery during the relaxation period better than the one I posted above, and in greater details. This time it is the full version, not only an abstract. If you do not want to read the entire paper, look at the Fig.2 and read the couple of paragraphs above it. I am attaching also a link of the standalone diagram below the link of the PDF. It shows the voltage recovery after the load is removed. The top diagram is for initially fully charged battery; the bottom one if for a depleted battery. The ideal recovery time at Li-ion batteries is around 2 hours (that comes from another paper)

http://www.ti.com/lit/ml/slyp086/slyp086.pdf

http://m.eet.com/media/1053387/TI_Algorithms_Fig2.gif

Update:

16.02.16    18:30:00    26000
    18:40:00    28800
    18:50:00    28700
    19:00:00    29000
    19:10:00    28900
    19:20:00    28800
    19:30:00    28700
    19:40:00    28600
    19:50:00    28200
    20:00:00    27500
    20:10:00    26200
    20:20:00    24200
    20:30:00    20000
    20:40:00    11400
       
17.02.16    18:30:00    13800
    18:35:00    10000
    18:40:00    5100

No significant recharging. 

I still have the contact problems, and the lamp was in a horizontal position so maybe the accus were not properly in contact with each otherduring the recharging time (maybe this is necessary for recharging ?) so i will give it another try by putting the lamp in an upright position until tomorrow so that the accus are pressed together by their own weight.

Visually the lamp still emits enough light to appear quite bright.
This shows that it is not possible to 'measure' light intensity visually, with the human eye, or with a normal video camera.
To do tests, it's absolutely necessary to *measure* the light output with a light sensor.

Regarding the insufficient contact - if there was no contact between the batteries when laying horizontally, the flashlight would not light, or would be blinking. It does not seem to be the case with your tests, as far as I understood.

... and BTW, Mr. Ivchenko of ADGEX has clearly shown that the batteries "recharge" when the switch is turned off, so it is clear that they do not need any closed circuit to "recharge", and hence any contact problems could not influence the "recharging"

Regarding the insufficient contact - if there was no contact between the batteries when laying horizontally, the flashlight would not light, or would be blinking. It does not seem to be the case with your tests, as far as I understood.

At the beginning of each test i did some shaking to get the best possible contact.
So it might be possible that during the recharging time there was no good contact for some time.

... and BTW, Mr. Ivchenko of ADGEX has clearly shown that the batteries "recharge" when the switch is turned off, so it is clear that they do not need any closed circuit to "recharge", and hence any contact problems could not influence the "recharging"

From a conventional point of view you are right, but from this point of view it would not work anyway.
So *if* it works, there has some kind of 'magic' to be involved, which we don't understand. 

Someone asked about the diameter of the bright light spot:  It has a diameter of 9 cm at 1 m distance from the front window.