I suggest gentleness and patience. Streaks after cleaning are usually the dried residue of the cleaner. (To clean I use a very low concentration of plain dish detergent.)
Alternate very soft damp (plain water only) and dry cloths, and stroke (not rub) the surface. Repeat several times. The smudges will gradually be absorbed. The screen coating naturally rejects contaminants, but it takes gradual encouragement, and is easily damaged by impatience.