It's a broken activation switch....contact C.S. The fact that the LED blinks 3 times instead of staying on and the LED light triggers upon taking a draw means the relay that sends power to the Atomizer from the battery is toast. This could be caused by a faulty relay or it could be caused by a loose ground wire.
Add edit: After watching the video a second time, the LED first blinks 3 times, then after being backed out of the charger it blinks 10 times. That would indicate that the battery is fully drained or is under 3.5v +/- which will still have enough output to power the LED light. You noted that you charged the battery or at least indicated that the battery was put on a charger and appeared to take a full charge. How long was it on the charger? Did you also check the charger contacts to see if they were clean and does it work fine with your other batteries?
Using prefills alone does not ensure that juice cannot get into a battery because they are directly connected to the Cartomizer. Batteries still get dirty, threads still get blackened for prefill users since the Cartomizer holds a little bit of vapor inside of it. Twist up a little rat tail with a napkin or paper towel and insert it into the battery post hole, keep twisting it slowly in one direction, it should go in about a half inch. After you pull it out and you'll see what I mean.
In any case you did everything that you could do.....