I'm using the same resistance as I was for the LED, 220ohms. Is that sufficient?
5 volts / 220 ohms * 1000 milliamps per amp = 22.7 milliamps. Which is less than 40 mA (damage occurs) so that is a reasonable choice.
Should the Mosfet have a resistor between the Arduino and white wire also?
I rarely include one but, yes, it should. The goal is the same for a MOSFET (limit the current). Someone else will have to help you determine a reasonable resistance for the MOSFET you are using.