Hi,
As you've learned, common tools like protractors are next to worthless. To be able to measure that small, you need to magnify the part you want to measure and then you need the correct scales. I personally would use an optical comparator. An inspection tool that can be had for $2000 to $10,000, depending on how big and fancy. And even then it wouldn't mean a whole lot as even new from the factory, the angle will change from run to run. Even shift to shift and operator to operator.
So, don't get so hung up on exact angles. There really isn't that much difference you can notice in the real world. No commercial sharpener ever made will provide for an exact angle. If they can hit within a couple of degrees either side for the marked angle, consider yourself lucky. Well, I could make it happen but it would require a surface grinder, sine bar, and a set of gage blocks, (and it STILL wouldn't be perfect. Just really, really close). And when you start free-handing, exact angles will mean even less.
If you want to match the factory angle, use the black marker technique as mentioned above. It's fast and easy to do. If you don't like angle, sharpen shallower for easier slicing, but less strength, or more obtuse for a stronger but not as good slicing edge. Experience will teach you what is best for you. Since we all use our knives a bit differently.
So relax, spend less time obsessing, and more time trying. And soon you'll have your knives so sharp they'll cut you if you just look at them wrong.
dalee