The battery can provide 20mA, that's right. But that is just one side of the equation. The current flow depends on the load, the LED in this case, which only needs about that 1mA (these candle LEDs have a built-in resistor).

The battery output depends on the anode,cathode material and the electrolite between them. You use some uncontrolled composition soil so the results are uncontrolled too.

Your soil has a neutral PH, which won't do much work. You need a bit more agressive stuff, maybe add a little acidic solution (vinegar for start)

And use longer screws, so they go deeper into the body. The larger contact area means larger currents (and larger current will deplete the battery faster).

And I forgot the flux stuff yesterday. Solve zinc in muriatic acid for a diy flux. I usually peel off tha casing of a used non-alkaline AA battery. You will need as many zinc as the acid can solve.

AND DON'T INHALE THE GASES WHEN YOU SOLDER!