I have the repair procedure for the washer circuit. The first thing you have to do is buy a 12 volt test lamp. They're available at most auto parts stores or in the automotive tool section of Sears hardware.
The diagnosis and repair procedure is as follows:
1. Check the washer system for water in reservoir, pinched/kinked hoses, loose hose attachments, plugged screen, or plugged nozzles. If there are problems repair accordingly, otherwise continue. (Note: the following steps require the ignition switch to be in the run position)
2. Backprobe WIPER/WASHER ASSEMBLY (which in the center of the firewall, in the engine compartment) connector C2 (which is connector on the front of the assembly) with a test lamp from cavity B (they should be in alphabetical order from left to right) to chassis ground. Turn ignition switch to run. If the test lamp doesn't light, the white wire that runs from the fuse block to the assembly is broken. If the test lamp lights, continue
3. Disconnect connector C2. Disconnect WIPER/WASHER CONTROL connector (which is the switch on the dash, and is a bitch to get out). Connect a fused jumper (you might have to make this) from connector the wiper/washer control connector cavity C to chassis ground. connect a test lamp from connector C2 cavity B to C2 cavity A. If the test lamp does not light, the pink wire that runs from the wiper/washer assembly to the control connector is broken. If the lamp lights, continue.
4. Connect a test lamp to from wiper/washer assembly connector C2 cavity B to wiper/washer control connector cavity B. If the test lamp doesn't light, repair the black ground wire for the wiper/washer control connector. If the light illuminates, reconnect the assembly connector C2 and continue.
5. Connect a fused jumper from wiper/washer control connector cavity C to chassis ground. If the washers operate, replace the wiper/washer control. If the washers don't operate, reconnect the control connector and continue.
6. Backprobe wiper/washer connector C2 with a test lamp from cavity A to chassis ground. If the lamp does not light, replace washer relay. If the lamp lights, continue.
7. Remove the washer pump cover. Turn wiper/washer control to LO. Observe ratchet pawl. If the ratchet pawl does not move, replace the washer pump assembly. If it does move, continue.
8. Press the washer switch and check ratchet gear operation. If the ratchet gear rotates, replace the valve assembly and retest system, if problem still exists, replace pistons and pump assembly. If the ratchet gear doesn't rotate, check for ratchet pawl spring missing or not properly connected, ratchet dog spring not engaging ratchet gear teeth.
9. Disconnect wiper/washer assembly connector C2. Disconnect wiper/washer control connector. connect a test lamp from C2 cavity B to cavity A. If the lamp lights, there is a short to ground in the pink wire, mentioned above. If the lamp doesn't light, leave it connected and continue.
10. Reconnect the control connector. DO NOT press the washer switch. If the lamp lights, replace the wiper washer control. If the lamp doesn't light, reconnect C2 and continue.
11. Remove washer pump cover. Check position of the armature leg. If the armature leg has lipped off the ratchet gear, check for dog spring not engaging ratchet gear teeth, worn ratchet gear teeth, worn ratchet gear spring tooth, and ratchet pawl spring missing or not properly installed. Replace parts as required. If the armature leg is in the correct position, check for loose pole pieces on washer relay or burr on end armature leg. replace parts as required.
I highly recommend to you and anyone who is planning on repairing their Cadillac themselves, that they acquire the factory service manual.
Here are some on eBay for an 92 brougham. I got all of this information from my OEM service manual, for an 89 Brougham, so some of the wire colors may be different.