Water ice in comets: Infrared/Microwave spectroscopes on a number of the spacecraft sent to comets all show large amounts of CO2 and H2O in the vicinity of the nucleus. Look for results for the Microwave instrument aboard Rosetta, and the infrared spectrograph aboard EPOXI. Production rates of water byproducts like OH- have been monitored for decades in various parts of the spectrum (but you appear to be more interested in direct proof of a H2O parent, so for that we need to look at spacecraft observations).
Comet Encke and sunspots : I am a bit dubious of the graph and I'll tell you why (Birkland was brilliant, but not infallible). If that chart were the brightness of Comet Encke, it would show a very dramatic peak every 3.3 years coinciding with it's perihelion. During the 1800's Comet Encke was really only observed for a window of 2 months around perihelion, and any time outside it would be invisible. Since Birkland's time we have access to far more observations and more accurate observations and I haven't heard of any correlation. However, feel free to do your own data analysis, there is a good modern database of comet observations here. Keep in mind that comet brightness observations are notoriously difficult and there can be a lot of scatter, particularly between different observers.
Those readings in the vicinity of the nucleus could be the product of an electrochemical process which would be most active on the nose of the nucleus where solar wind(sheet current) is in greatest contact. Of the four comets that I am aware of that they have gotten close to there seems to be very little to no presence of water. The fact that there is this discrepancy between what we can see in the visual light range and what the sensors are reading suggest that there is something more to this. We agree that there is the presence of water, what we do not agree on is how that water is present. You insist on snow or ice, even though the images do not show any. Oh yes, it is subsurface, but even then from what I can ascertain, there was none observed by sensors or otherwise when they impacted a projectile and analyzed the ejecta. While I insist it is being produced via electrochemical process stripping atoms off the rock.
True enough about the brightness being difficult to measure and compare. Thanks for the link, I will check it out and given enough time may even link up a database on sunspots and graph the two together for comparison. A single comet clearly does not define all comets. But I do think Birkeland was on to something there.