Cleaning the stick blender is so easy - just add hot water to your sink or your soap batter bowl. Then run the blender at full speed for a minute or two. The dried up soap just blends away into the hot water.(I HATE cleaning the stick blender lol!)
If you need to clean the blender before the soap on it has dried (because you made one batch and want to make more right away, haha), add some grease-cutting dish soap to the hot water before blending.
I do the same with my Vitamix after making smoothies - hot water, few drops of dish liquid, and blend for 30 seconds. Works like a charm