I looked at the 555 circuit and it is supposed to be on for about 750 us, then off for about 33 us, so the pulses where inductor is on are only 33us in duration, during which time the inductor's current should increase only to about 400mA. In theory, this should be fine. So, 5A rating that I suggested earlier might be an overkill and the inductor rated for 600mA - 1A should be fine, but the 100mA inductor would definitely saturate. If it saturates, you get high currents and something will go.
Another question is whether you get the 95% duty cycle that it is supposed to produce at pin 3 or not? Can you measure the duty cycle?
As a quick fix, I would try decreasing R2 to 10K and then further to 4.7K. This would make "on" pulses shorter while keeping the frequiency, so you can get away with smaller inductor.
I wouldn't change the capacitor. It's probably chosen to set the frequency believed to be good for the desulphation.