After a few days some news:
I couldn't find out, why your code isn't working, Paul. I think it has something to do with the sequence the MIC5821/MIC5891 work.
I post a excerpt from the Datasheet:
Serial data present at the input is transferred into the shift register on the rising edge of the CLOCK input pulse. Additional CLOCK pulses shift data information towards the SERIAL DATA OUTPUT. The serial data must appear at the input prior to the rising edge of the CLOCK input waveform.
The 8 bits present in the shift register are transferred to the respective latches when the STROBE is high (serial-to-parallel conversion). The latches will continue to accept new data as long as the STROBE is held high. Most applications where the latching feature is not used (STROBE tied high) require the OUTPUT ENABLE input to be high during serial data entry.
Outputs are active (controlled by the latch state) when the OUTPUT ENABLE is low. All Outputs are low (disabled) when the OUTPUT ENABLE is high. OUTPUT ENABLE does not affect the data in the shift register or latch.
so maybe the digitalWrite(CLK, LOW); isn't working and the digitalWrite(LATCH,HIGH); must be executed before the data shifting starts. But it is only a assumption by me...
Nevertheless I'm going to try to unse a timer interrupt to shift the data and set the OE and LATCH pin. Maybe this will help.
Meanwhile maybe there are some other ideas, which might help me? Any advices are appreciated! Thanks! 