In your previous thread, you mentioned your inside chamber dimensions are 11"x4.5"x3.75", for a volume of approximately 0.1 cubic foot. That is a pretty small volume and you can easily run that on 110V. Commercial ovens typically have 4-8kW per cubic foot, an order of magnitude less than 15A at 240V for your chamber volume. For that size, 500-1,000W should be plenty.
While JT is correct in everything he says above, he does not address how to fit the coil into your furnace. The larger the wire gauge you use, the more wire is needed for the same resistance. You have some limitations and my guess is that you are running the coil back and forth in two channels per side that are sized to accommodate a 40" long coil with a diameter of just above 3/8". If you do not want to add new grooves, your options are pretty limited. If you can widen the grooves, you can fit a bigger coil, i.e., more wire length and thus a thicker gauge wire. You would have to use a 1/2" mandrel to wind your coil if you wanted to use 16 gauge Kanthal at around 40" stretched length to get to the power range mentioned above.
My recommendation would be to rewire for 110V and use an 18 gauge coil and see how long it lasts. When it burns out, you might want to build a bigger furnace anyway (I do), or add new grooves to accommodate a longer coil of thicker gauge. If you have a multimeter, measure the resistance of your current coil, or maybe let us know how many feet of wire you used to wind the coil. You might be able to just use that on 110V and it could last a good long while. If you want to make a new coil from 18 gauge Kanthal, I would recommend you start with 32-35ft for about 16-18 Ohms and 800-900W. You can go a bit shorter for more power (and reduced lifespan), but I would not recommend going much longer as that will prevent you from sufficiently stretching the coil.