Uhhhhhhhhhhh............
You could always just go analog and use a potentiometer, only using the Nano to monitor the temperature and give you a serial readout, if you even need it at all. That way you don't even run the risk of frying anything, unless you accidentally smoke a pot meant for a lower amount of power.
But as far as I know, you seem to have the right idea, using a mosfet and PWM. All I would worry about is making sure you have a separate logic power source and a resistor between the mosfet output and GND on the Nano to avoid current spikes. Wiring is ket in these sorts of setups. That and having at least a basic idea of what your code does.
My advice? Go simple. Start with a baby code, then work your way up to the complex stuff like temp monitoring and such. Secure the basic functions before adding extras. That means get rid of the temperature sensor until you know the hot end works like you want it to.
And also, as I said, power the soldering iron bit and the Nano with different batteries. You will need to implement a common Ground on the Mosfet, but otherwise the wiring is pretty much the same. And make sure on the Nano you are putting volts into the Vin pin, not the 5V pin.
Anyway, I think thats everything helpful I can offer. A picture of the actual circuit might bring a better understanding as to why it won't work properly.
-D.E.