hmm, im not sure but it looks like you have crap IN your screen, not ON your screen... that happens when you crack your screen, and the liquid crystal stuff inside "spills" all over the place inside - there is no cleaner on earth that will fix that =(
If its stuff on the screen... ive been using windex (and the like) on my screens all my life, and have seen no negative effects from doing this.
Just make sure that your screen is cold when you do it, and dont use alot of spray, and use a GOOD towel - a T-shirt is actually one of the best things next to a real micro-fiber cloth to use for that.