The white film is a calcium and mineral deposit from our water. The same as in your teakettle or around your faucets if you're in an area with hard water...(AKA: So.California).
You can try a mix of elbow grease and:
Product #1: http://www.oxyboost.com/productPages/mineralMagic.html
Product #2: http://www.intlstoneworks.com/specialty/mdr_instructions.htm
Product #3: http://cmcchemical.com/featured/210_mineral_deposit_remover.html
Product #4: http://www.scumoff.com/
With or without any of these products, you can use a single edge razor blade scraper with handle and, after applying a soapy film to lubricate the surface, lightly scrape the worst areas with the blade flat against the surface at about a 45 degree angle to the surface in the direction of the push.
Then use something like a 3M plastic scrub pad (usually green) and an everyday scouring product such as "Bar Keepers Friend" or a "soft scrub" scouring powder. Again, don't scrub with a lot of pressure, just use a lot of back and forth scrubbing with light pressure and keep checking the surface by wiping it clean with a sponge and dry with a towel to see if the film is gone.
After hard water scale is removed, apply a liquid polymer car wax. This will inhibit future accumulation and lessen the effort to remove build-up.