Step 1: determine how much heat is lost. If you system is very steady this can be easily determined from the temperature change, but if you have additional information about relatively fast changing factor (changing environmental temperature, wind chill factor, cold stuff being added to the mix, etc) taking that into account will give you a much better performance (accurate temperature control) than the most sofisticated control that is only based on the mix temperature.
Step 2: design your control. I am no PID expert, others are. Output of this step is the amount of heat you want to put into your system at each moment.
Step 3: the circuit that controls the heat that is put into the system. I very much doubt that you need this heat flow to be accurate on a sub-seconds base, likely it will be OK when it is on average correct on a 10 seconds or even minutes timescale. Hence you can switch individual half-sines (0-cross switching), which is far easier (and produces less line pollution) than phase cutting (is that the correct term?).
If you need fine grained control of the amount of heat you put into the system you could
- measure the voltage (current is probably not needed because your heater will be resistive) with sufficient time-resolution to calculate the total heat produced by each half-sine that you pass through to the heater
- use delta modulation to determine whether to switch each half-sine on or off