Diving a metal rod into ground is not an option because the apartment is on the 10th floor 
Anyway, I've just tried the touch-the-LED-touch-the-grounding experiment in my parents' house. Here everything is fine (I got a quick pulse of light from the LED on first touch but nothing ever since - I guess there might have been some static electricity left in me or in the iron), so I decided to move the LED project here. I don't really have another choice. The grounding in the apartment where I've started the project is giving off current at 90 V, so it's dangerous for sensitive equipment 
Good thing I managed to figure it out in time. I've only wasted 12 hours of work, about 70 LEDs and about 6 meters of good craft wire. Better like this than being sorry with the 8x8x8 cube.
Thank you all for helping me figure this out! I really appreciate your help.