Some thoughts...
1) In the video, you removed 2 LED's. Did you test the light after you did that? If you just pulled the LED's, you may have left an open circuit and the light will no longer function. If you just connected the two leads that went to the LED, you will likely be overdriving some or all of the remaining LED's and cause premature failure.
2) An AA/AAA battery will be about 1.5 volts when charged, and drop to more like 1.2 as it discharges. So that light will run on about 3.5 to 4.5 volts. If you put 12 volts into it, you will almost certainly destroy it very quickly considering it just uses a resistor for current limiting. Based on V=IR, the more volts you put in with the value of R being the same, the higher the value of I goes.
3) The suggestion about to drive them in strings of three - that does not limit you to only three lights. You can have as many strings as your power supply will support, but each string will be 3 in series. That would give a forward voltage of 4 volts to each light, which would have them running as if they were on not-full-but-not-empty batteries.
4) As said above, make SURE your power supply is 12VDC. Many 12V transformers are not rectified, so they take mains voltage and output 12 volts AC, which won't work... well, there's a bit more to it than that, but suffice it to say you want to run these on 12 volts DC.
To wire 3 lights in series... if you look at your light, despite having 3 batteries, there will just be 2 wires or connections between that battery pack and the LED board... positive and negative. So you would connect the positive from one LED module to the negative of another. Then connect the positive of that second LED module to the negative of a third. Now you have 3 in a string (in series) with one leftover positive and one negative connection. It will require 10.5 to 13.5 volts to run that string.... which is right in line with what your 12 volt DC power supply will provide.
You can have as many strings as your power supply will handle... simply connect the aforementioned positive of a string to the positive of another string - and connect to the positive of your supply. Then do the same with the negatives... and you can have as many strings of 3 as you like.
Always do strings of 3 though...any less will cause them to burn out prematurely.
The right way to do this is to use an LED power supply that provides constant current, but it sounds like you're looking to do this fast and cheap- so that's likely out of the picture?