Go to an office supply store and purchase a couple of items: graffiti remover/marker remover for white boards, and white board cleaner.
Secondly, I would see if the same store sells microfibre cleaning cloth and use that cloth for cleaning your screen while using these items. If the store doesn't have it, just go by an automotive store and ask for a "detailer's cloth" which typically is microfibre. Finally, if that doesn't work go by an optometrist and purchase a lens cleaning cloth.
I would first apply the whiteboard cleaner to the cloth, then experiment on an out-of-the-way corner, just to make sure the material and cloth is safe on your screen. I'm sure it would be. Then venture to the permanent marker area and work with that.
If the white board cleaner doesn't work, then get the graffiti remover and give a try with it. Experiement in a screen corner first.
Hopefully, this will work and you are not using anything that is abrasive, also these chemicals are known to work well and safely with plastics but I would still work with a corner first, as a test.