Hello again!
I did some more tests and... I'm even more confused 
First of all, I measured the voltage between the tip of the soldering iron and the wall socket's grounding. I've set the multimeter to 750 V AC and tried it. The result: 90 V! Wow! I know that there's no error in my measurement because I tried measuring the mains voltage with the very same settings and I got 230 V, which is the standard value here in Eastern Europe.
Next I took another old soldering iron, which does not have grounding. With that one I obtained 45 V.
You know that what I initially observed, yesterday, was that if I touch either leg of the LED with the tip of the iron and the other leg with my finger, the LED lit up (although it was dimmer than usual). So I thought to myself, let's try to ground the tip of the iron by connecting it with a wire to the wall socket's grounding. And this is when the real surprise came! The theory was that the current will not go through the LED, it will go through the wire directly to the wall grounding and the LED will not light up. Instead the LED lit up twice as bright! Remember that I was still touching the LED's other leg with my finger.
Next I completely took out the soldering iron from the equation. I just held one LED leg with my fingers and I connected the other LED leg directly to the wall grounding through a wire. The LED still lights up, although not as strongly as through the soldering iron.
So what is going on here? Am I producing electricity and it's flowing through the LED into the wall grounding? And for minutes and minutes, without an end? How can I be producing electricity constantly? OR is the electricity coming from the wall grounding and going into me?
I tried holding the LED and connecting the other leg to some other mass, like the radiator, that way it does not light. But if I connect one leg to myself and the other leg to ANY wall grounding in the apartment, it lights.
What's going on here? The only thing I can think of is that the wall grounding is not done properly and that it's leaking very small amounts of current at 90 V...
I'll try taking the LED into another home in a different part of town, let's see what happens.
What a stupid game of circumstances that lead to the destruction of my 4x4x4 cube!