Probably the same routine that I use to repair a fine wood carving gouge which has hit a sand grain.
I prefer to do all this freehand as I've never gained much experience with powered sharpening systems of any kind.
1. Paint the existing bevel with black felt marker so that you can see what's happening.
2. Begin with a coarse oil stone. I use Pennzoil 5W40 motor oil. Hold the tool axis vertical and scrub the tip back to square with the axis. Tedious but has to be done once.
3. Draw the desired bevel angle on some sort of card stock. This will stand beside the sharpening medium until the end.
4. Put the old bevel tip on that same stone, same oil, and elevate the tool shank to match the angle card.
5. Scrub and scrub to re-establish the needed bevel. I have skews at 25 degrees, spokeshaves at 28 degrees and my Stanley #5 at 30 degrees. I like those angles. Most western wood carving tools are 20 degrees. For a mortise chisel, that may not put enough steel behind the edge for support in that task.
6. I'd go through my 3 oil stones refining the bevel, the top might be approx 220 grit (unknown, all of them.)
7. 600, 800 then 1500 grit W&D automotive finishing sand papers.
8. From step 5 onwards, you'll notice a good wire edge develop. Alternate a few strokes on the flat, maybe pull strokes only. Takes time but it will break off.
9. How fine you need to go is up to you. Who sees the inside surfaces of a well cut mortice?
10. I don't like sanding my carvings. As such, then, the tools mark the surface. I hone my edges with chrome green on some sort of strop = might be leather, might be crackerbox cardboard, might be a 4" x 6" filing card. With a nominal particle size of 0.5 micrometer, that makes my honing process about 40,000 grit.
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Looking again at the tip, it is so very rounded off, is there a possible reason for that shape, not just simply wear and tear?