You really need a base resistor. 0.3A says that (most likely) there is WAY TOO MUCH base curret! Only a small percentage of your power is going into the LED.

I changed the wires in the toroid for telephone wires with only 5 turns and assembled it as before and it finally worked!

I managed to lit it with a battery with 1 volt left, but for some reason it does the flash with the other batteries although there are ones that are not discharged. With an AA battery the transistor got hot. Probably I need to get a higher value resistor.

Congratulations for getting it working!

(Rhetorical question) Why do you measure 0.6 volts across the LED?
Take a look at the circuit. Notice that there is a direct current connection between the batter + terminal and the anode of the LED. The average voltage on both ends of an inductor are the same. That also tells us that the battery voltage at the time of the measurement was 0.6 volts.

The peak voltage across the LED is not DC, it looks like the waveform shown below (image harvested from the internet).



The peak voltage across the LED is about 3.5 volts and the duty cycle is about 40% which tells us the battery voltage minus a little bit lost in the transistor is about 3.4 x 0.4 = 1.4 volts.

Something to keep in mind - at DC inductors look like shorts and capacitors look like open circuits. At high frequencies inductors look like open circuits and capacitors look like shorts. In the region between low and high frequencies the components have reactancces somewhere between zero and infinity.

 

I don't know what happened but when I connected the same battery that I used yesterday the JT stopped working. Did I somehow ''disturbed'' the oscillation? It happened the same way as when I used the telephone cables.

 

I fixed it again. Apparently the cause of error are the connections, winding issues, or maybe the transistor needs ''cool off''.