Tom, if the fuses are OK, it could be the wiper/washer relay. Does the windshield wiper/washer work correctly? They share some common circuitry but have separate pumps. On a W124, I'm unsure of a W210, there are two pumps mounted on front side of the reservoir. The lower is for the headlamps; the upper for the windshield. Check the push-on electrical connectors there.
It wasn't clear from your post if the headlamp WIPERS were working or not
And you probably know this, but at least the parking lamps have to on to activate the headlamp wiper/washers when pressing the cleaner switch on your stalk. If you can find the sqare black wiper/washer connector (somewhere around or under the headlamp assembly) and the wires which go into it, two leads (brown-neg, black-pos) should have +12 V whenever the lights are on. The red lead should get a momentary +12 VDC when you press the cleaner stalk. And only on the RH side, the blue lead supplies +12 VDC to the pump at some intervals while the wiper makes its cycle, shutting off when the wiper does.
If it's a wiper/washer assembly problem, one cause is that moisture will get into the wiper motors and over time, if the lubricant inside is inadequate, will corrode and cause it to freeze up. Wear can certainly degrade the components as well and cause misoperation. If this is the case and you want to pursue a repair rather than replacement, see the detailed treatise at http://www.qadas.com/~kanas/300TE_Headlight_Wipers.html