some quick thoughts
Once you have drilled all the way through, how would you know at what point the bit hit water? Also, would need quite a long bit -- 2 ft of ice is not unusual. I suppose if you just wanted to test whether the ice was at least, say, 5 inches, you could use a shorter bit, mark the bit with tape or paint at the desired depth and drill to that level. If no water is found, then you know it's at least that thick. The problem is that you won't know how "good" the ice is using that technique -- meaning how much is slush ice vs. black ice. Just seems hard to get away from a handy manual ice auger -- might only need a 4" to be able to see where the black ice begins. Then I use a marked stick with a nail sticking out at 90 degrees to hook the bottom edge of the ice and measure up.
Google the Nimrod Ice Auger adaptor. It let's you use a cordless drill to connect to a number of brands of manual auger bits. Not sure how well it would work in extreme temps.
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