As the owner/operator for 30+ years of my own domestic services business, may I suggest this tried & true solution? Don't use any "wet" products. Instead, take a 0000 steel wool pad and with minimal effort, simply lightly scrub the dry residue away. Your glass should absolutely gleam when through.
I've rarely, if ever needed to use anything other than this very fine grade of steel wool for glass, chrome & mirrors of all kinds.
Vinegar & Newspaper - mix 1 part vinegar with 1 part water into a spray bottle.
Spray the effected area, leave for a minute then wipe dry with a sheet of newspaper. You may have to repeat this several times, if the residue has been there for some time, but be patient. A little elbow grease can save you a lot of $$.
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