The Ballast resistor is just a low resistance in series with the motor. You can tell if it's working if you have a multimeter.
I dont have the board in my hand so my description of how to measure it is a guide.
Either end of the long curly track on the green board will probably go to the pins on the connector, just trace it through visually. The little W thing is prob a safety fuse in series with the long curly bit, (I'm guessing please see caveat at end). When you've found the pins it goes to, with the multimeter switched to resistance or ohms ***937; touch the probes of the meter to each other and the meter should read around zero ish. Now put the probes on the pins connected to the track and you should get a reading of a few ohms, I don't know the exact figure but it doesn't matter, could be anywhere between A fraction of ***937; and 20***937;. If you get any low resistance reading and the continuity of the track is ok then its prob working ok, if the meter reads open circuit or very high resistance it's not ok. Get someone who knows how to use a meter if you aren't sure.
There are safety implications in replacing the sections of curly bit with (wirewound) resistors, you can't do that unless
1) the W safety link is intact,
and 2)the wattage rating of the resistors is sufficient as they will get hot
It's probable that if it has failed either the track will be cracked near the connector or the safety W will be burnt out. (I'm only guessing that its a safety W, it might just be another low resistance to control the speed?)
hope that helps a little.