Okay, so it's running from 3 x AA = 4.5V at most, so shouldn't have any series of LEDs, all in parallel.
If it was driving the LEDs too hot, it is definitely possible to blow them, but normally what would happen is those with the lowest (will vary slightly from one specimen to the next LED) forward voltage will blow first, so initially you would have one, then a second, and so on LEDs failing, not all at once.
All at once suggests to me that it's more likely you have a cracked resistor solder joint or something as this kind of cheap (R115 = $8USD?) LED light often skimps on resistors barely rated for high enough wattage if that.
As for why your testing isn't getting it to work, with low resistance,I can't see the test and can't see the circuit. There is no reason why the circuit can't be completely pictured without being assembled is there? That should have been the first thing done so that we can clearly see it.
Is it possible that the 12VDC external power socket is damaged, that it has a switch mechanism in it to switch between internal battery and 12V external power, and it is shorting out?