I've used a number of different brands, Dove, Palmolive etc. They all seem to work well. Had mine for 2 years now.
Just remember to use a very little bit of the detergent, especially with anything labeled concentrated - don't over do it.
I usually mix it with water in a bowl then clean the cover with a wash cloth or just my soapy hand.
I also make sure it's super dry before I put it back on but I guess that goes without saying. 🙂