What resistors do I need
The micro-controller will output near to 5V as that is what goes into the VCC pin.
As to "..using the formula" , I don't know what formula you have been given.
R2 depends on what type of LED you have. 1K seems a safe value which gives ~4mA through the LED.

R1 can be any value between 100 and 1K. 

I would add a 100K resistor from the FET gate to ground to make sure the FET is off when the micro controller is still booting. (And whilst you are developing the software). 

I had a quick look at the WWW and most pyrography supplies have a control knob which seems to be missing here. I assume you want to add one because what you have here does not requires a micro controller at all. You can do the same with just a switch.