Diagnosis
You're right, there's no dedicated 'clean' button. The cleaning process is manual but follows a specific sequence outlines poorly.
How to Fix It
Here's the efficient method. Power off. Empty all ice. Fill reservoir with a 50/50 mix of white vinegar and water to the MAX line. Power on and let it run a full cycle until it harvests vinegar-ice (about 15 mins). Dump that ice. Now fill with fresh water only and let it run 2-3 cycles, dumping ice each time, to rinse. Wipe the exterior and ice basket with a damp cloth.
If That Didn't Work
If scale is heavy, use a commercial descaling solution made for coffee/ice makers instead of vinegar, following the same process. For exterior mineral spots, a paste of lemon juice and baking soda works.
Heads Up
Do not use abrasive cleaners or scrubbers on the interior ice mold. You'll scratch the surface, causing ice to stick permanently. Do this cleaning routine every 2-4 weeks depending on your water hardness.