It goes without saying that electric kettles should not and should not be immersed in water, so the possibility of washing kettles in a dishwasher is excluded. At first glance, it seems that kettles do not need cleaning, especially if you traditionally only use boiling water.
On the other hand, some non-electric kettles are safe enough to be dishwasher safe. Stainless steel kettles are dishwasher safe, but you need to check with the manufacturer to ensure they are safe.
Therefore, it is best not to wash the kettle in the dishwasher. Do not immerse the kettle in water, as this will damage the heating element.