That LED will probably burn out immediately. The data sheet for this typical red through-hole LED lists a max forward voltage of 2.6 V and a max current of 20 mA -- your circuit will probably give it upwards of 11 V and significantly more current. Check the limits for your momentary button, too. Since you have a MOSFET, so you could use a reduced voltage & current for the button, LED, and to control when the MOSFET should let the full voltage & current to the induction heater.

I'm not familiar with that battery charger unit. You might want to connect things without the batteries, running directly off the power adapter, and make sure things work that first.