DIAGNOSIS You're right, there isn't a dedicated "Clean" button. The cleaning process is a manual one you initiate by holding down a specific combination of buttons.
FIX Here's the shop-teacher method. 1) Ensure the ice basket is empty and the water tank is filled with a cleaning solution—use either the official Newair cleaner or a mix of 16oz warm water and 2oz distilled white vinegar. 2) Press and HOLD the "Ice Size" and "Power" buttons simultaneously for about 5 seconds. All the indicator lights should blink. 3) The cleaning cycle will now run for about 8 minutes. It will pump the solution through the system and then drain it. 4) When it stops, dump the leftover solution from the tank, rinse the tank thoroughly, refill with clean water, and run a normal ice-making cycle (discard this first batch of ice).
IF THAT DIDN'T WORK If the button combo doesn't trigger the blinking lights, double-check that the unit is on and the lid is securely closed. If it still won't start, you can do a manual clean by unplugging the unit, removing as much water as you can, and using a long-handled soft brush and vinegar solution to gently scrub the interior water channels and the evaporator plate you can see.
HEADS UP Never use abrasive cleaners, bleach, or harsh chemicals. They can damage the plastic and silicone inside and leave residues that get into your ice. Stick to vinegar or citric-acid-based cleaners meant for appliances.