You need to open the detailer and check the cord connection to the circuit board. It just may be a loose connection that has to be resoldered.
If it looks OK, then if you have a DMM (Digital Multimeter - Ohmmeter function) you need to test the continuity of the wires from the connection on the circuit board in the detailer back to the plug. The way to do this is with your meter, connect one meter lead to one plug connection and then find out which wire it is by connecting to one or the other cable connections in the detailer to get a reading in the meter. Once you have established which wire goes where between plug and detailer, (do it for both wires) leave the meter connected so that it is reading a connection THEN gently flex the cable, if the reading doesn't change the wire is OK, try the other wire and do the same. If the reading disappears and comes back when you flex the cable you know which wire is faulty.
By holding the cable firmly and flexing it at different points to see if the meter changes or not you can take a guess, at how far into the cable the wire is broken and cut the cable at that point. Test the wires again to make sure that they are now OK and then re-terminate them back onto the detailer observing where the wire colours are. (take a picture before you unsolder the wires as a reminder).
It may be easier to replace the cable if you have to cut off too much to remove the faulty section or if you don't have access to a DMM.
Ensure that the detailer is NOT plugged into a power outlet at any stage during the test and repair process and that it is not plugged in until you have completely finished and replaced the detailer's cover. With the cover removed there are exposed lethal voltages inside the detailer when it is connected to a power outlet