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Just lap on water. You can use a hacksaw. Slate is soft. Last time I had a skate with a break like that, I used a coping saw of all things and it still got it done. Just lol the shard off to square...

It goes without saying, hack off the spar at the one end and gain yourself a slurry stone for further experimentation.

Pretty much anything you have would be fine, they aren't brutally hard and are compliant to lap.

It appears to be a mottled piece of Vermont slate. It could have been a hone or building material, but either way it is well worth testing. Lap it and test. In my experience, best not to use water....

Where are you located, it helps with this