I'm not sure if the Klima relay runs the tach. If your tach isn't working, it might be easiest to grab the instrument cluster out of another wrecked 126 body car and hook it up. If the tach works, it's ok, if not, it could be another problem...
My ac problem on my car was fixed by simply putting in a new klima relay. Testing mine was rather simple. I opened it up, and there was a place where the circuit board circuitry was melted. It was a big piece melted out too... From what I understand, the klima relay is usually the problem on these cars...
However, it could just be a sensor problem, or a wire broken or unhooked... That I dont' know how to check.

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'95 S320 LWB
'87 300SDL
'04 E500 wagon 4matic