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 off a bit, grid the surface with a pencil or marker, cancel the grid, grid it again, cancel that one, round off the corners give it a try. For future reference, was way to lap anything not brutally hard is to buy a single granite tile that was polished/lapped and just use that as a base for your wet dry sandpaper. Start at 220 work up to 600-1000 if you want. Anything really hard, you are best off with loose grit silicon carbide on float glass