Simple green. The force of the point blank spray action of the spray bottle should be good enough to get the gunk off. Then rinse with water. Rinse and repeat if necessary. You can rinse it with isopropyl alcohol for good measure (but not required). Let it air dry for a few days just in case before using it.
I wash entire monitors all the time with this method and I never remove the ring. They all come out fine and have worked for years afterwards.