I have used the shower room in the Business Class KE lounge on several occasions. The room will have soap and shampoo. If I recall correctly there is also a supply of wrapped toothbrushes containing a mini one use tube of toothpaste. These are the same toothbrushes available onboard KE flights in the restrooms.
The shower rooms I have used did not have a toilet. It was located in the same general area but not in the shower rooms I have used. I hope this is helpful