This works for cleaning a lot of things. I regularly clean the coffee maker (a Bialetti espresso thing) by putting some powder detergent in the water compartment, then „make coffee“ without actual coffee and leave it in overnight. Neutralize with white vinegar (which has the added benefit of descaling) and rinse thoroughly. It‘s as new afterwards.

And yes, it’s not good for your skin, so avoid immersing your hands in it or wear gloves.

I don't think boiling water with dishwasher detergent in it is a great idea (because of potential fumes). I might have misunderstood what you mean by "make coffee" but maybe something to look into if you do this regularly.