As Earl pointed out, you don't need the gate resistors. BUt, for the bi-polar application, here's how I come up with those values:
When the input voltage is at 10V, we want 5V at the heating element. For that to happen, the voltage at the base of the transistor needs to be around 5 + 1.4 = 6.4 [1.4Vbe].
I figured the Hfe of the darlington to be around 5000, so at 2 Amps, the base current [Ib] is going to be around 2Amps/5000 = 0.4ma. Set the current through the voltage divider (i.e. the resistors) to 0.4ma * 20 = 8ma (multiply it by 20 to swamp out the effect of the base current).
So, the lower resistor (the one between the base and ground) has 6.4 volts across it, so it's value is: R = 6.4V/8ma = 800. Nearest standard value is 820.
Which will but the divider current at: 6.4/820 = 7.8ma
Then, to get the upper resistor, subtract 6.4 from 10 = 3.6V, which is the voltage across the upper resistor. If we ignore the influence of the base current: R = 3.6V/7.8ma = 462
Now, lets try that with the expected base current and see how different it is: R = 3.6V/(7.8ma + 0.4ma) = 439, so I went with 420. But, perhaps 470 would be a better choice. It all comes down to what the base current will actually be, and that will depend on the Beta of the transistor.
I was using 5% standard values but you could also use 1% resistors and come closer to the calculated values. Also, if you up the ratio between the divider current and the base current (i.e. instead of 20 use say 50 or 100, etc), the base current will have even less influence on the divider voltage, BUT that will put more of a load on the PID output.
The advantage of using a MOSFET is there is no need to divide the PID output voltage -- thus no need for the dividing resistors. You could also achieve the same thing (i.e. eliminate the divider resistors -- replace them with a single current limiting resistor) by applying around 5.9 to 6.4 volts to the collector of the transistor (depends on what the Vce-sat is).
This just in: I took a look at a transistor selection guide (http://www.mouser.com/catalog/supplier/library/pdf/FairchildBipolarPower.pdf) and discovered that an Hfe of 1000 to 2000 is more realistic.