Sounds like when jumping klima you managed to get the magnetic coil that draws the clutch in to function likely if the engine was running the compressor plate(shaft) would have been now turning. You could at least verify function with engine running and blower running, diagnosing the sound or condition of the compressor and possible cooling before spending alot on the electrical components when expensive refrigerant components may need replaced also or first.

Unfortunately The functions of the safetys of the compressor electrically on theses cars through HIGH/LOW switch, Coolant Temp switch, CCU, Klima and Speed Sensor to name a few (possibly also evap coil low temp safety) is a foreign languauge to me and I could even try to explain it to you. Id assume someone who has troubleshot that portion before will chime in.

Good luck

Arlo

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W211 E550
HVAC/R Certified Technician