I'm late to this, but a great way to get hard water spots and "film" that you sometimes get on the cooking surface is Twinkle copper cleaner. I discovered this when cleaning the tarnish off some copper clad stainless saucepans and found that it easily and quickly removed all the stuff that was adhering to the inside of the pan as well. I could never get the stainless shiny before, and didn't want to scuff or scratch the smooth surface with any sort of harsh abrasive. This stuff works perfectly.
You, of course, have to make sure you wash the inside thoroughly after you rinse off the copper cleaner, to remove all the residual.
This trick also works on some pretty heavy-duty stains and buildup, and might actually have worked on the OP's pan. I often use a Dobie or other similar pad to help provide a little "dig" when working on anything that has some thickness to it.