PID controller - How is plant equation derived?
PID really isn't appropriate for soldering irons. A simple proportional loop with high gain is a much better investment. The big problem is that the iron works in two regimes, resting and loaded, and the loaded regime is very badly defined. 

Starting with the available heater power, the mass of the coil and tip, and the surface area of the hot section of the iron, you can get an idea of the temperature sensitivity of the iron. Thermal impedance from coil to air is the important factor, with the heat capacity providing another constraint on time to reach desired temperature.

A PID can be used to reach operating temperature in minimum time and maximum accuracy, but the working temperature is typically not extremely precise (a degree or so for extremely demanding applications, or 10 degrees for a "real life" soldering iron. 

The thing is, once you go to actually soldering, the amount of heat you need depends greatly on exactly what you're soldering. A big, heavy wire or buss bar will require much more heat than an IC lead, and the thermal lag in the iron tip will mean that response times of a few seconds can be expected, and will depend on the target.  

So, depending on what you're soldering, the plant model changes and you cannot optimally tune a PID. A simple proportional loop will work adequately for any likely real-world soldering iron.