Since your application (electric furnace for fused metal) is not going to need
quick reaction times, one flexible way to control the heating power is to use
PWM (Pulse Width Modulation), as mentioned by Neil_UK in a comment.
PWM can be generated by a small microcontroller and then used to drive the heating wire through a driver circuit using MOSFETs.
Given the thermal inertia of the wire-furnace system, the PWM signal could have a rather low frequency (in the hundreds of Hz range), easily generated by even the most basic MCU using just bit-banging, although using a higher frequency (above 20kHz) could increase efficiency of the switching MOSFET. There are tradeoffs in this choice.
Even if the MCU is just used as a PWM generator and not in a full closed-loop control system, it will allow you to set the power in a very simple way.
In fact you can hook up a simple potentiometer read by the internal ADC of the MCU and then convert the pot setting to a duty-cycle setting in order to set the desired power into the wire.
If you want to be more sophisticated, the MCU could be connected to a thermocouple that monitors the temperature of your furnace and the reading could be displayed, together with the preset duty-cycle, on a simple LCD module, also controlled by the MCU.
At this point, if you want even more sophistication, the system is easily modified (in software) to become a closed-control system, with the MCU automatically setting the duty-cycle of the PWM signal to obtain a preset temperature.
Since your application uses AC, this simple approach would need to convert the AC power to DC. This is easily done with just a rectifier diode and a cap (ripple is not an issue in such a circuit).
If DC conversion is not an option, you could use a TRIAC-based power controller using phase-control. These analog circuits are simple and reliable for powering resistive loads, and have been around for decades (the classic incandescent lamp dimmer circuit).
This latter solution doesn't involve an MCU. However, you could use an MCU to control the TRIAC and have all the benefits I mentioned before (LCD display, etc.).
Here is a quite comprehensive application note from ST about TRIAC control using an MCU in AC applications.