The voltage that comes into the assembly first goes to the pull-cord light switch via the black wire. More than likely the switch is bad. The output of the switch goes to the light fixture. If you have a voltmeter or a voltage tester, put one lead on the white wire and the other on the black wire going into the switch to make sure you have the voltage. Then put the lead on the wire leaving the switch. Operate the switch a few times, you should measure 110 volt on the input all the time and voltage on the output on alternate pulls of the cord.
Adios... Jasper