In terms of hinges, I had roughly the same problem as you do with an old cabinet I'm (still!) restoring. I was not happy with the existing loose/damaged ones so I removed them, cleaned up the hinge slots with chisels, and filled them in with some scrap maple (I made no attempt to match the color to the old wood). Then I installed piano hinges.
Built with just a few simple tools and inexpensive materials...and lots of desire....I think Mark did a fine job building this hand tool sharpener, especially true when all he had to go by was a few pictures and no measurements or prints..Good job, Mark!!!!
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