Small drills are really hard to sharpen without a fixture. I don't have one, so I buy new drills when the small ones go dull.
Drilling lock plates: Run as low as your drill press will go. Use a sharp drill and cutting oil. Not motor oil. Many cast steel lockplates are a medium carbon steel, which will work harden if a dull drill or a drill that has been run too fast. The hole can be so hard that you need a carbide bit or the plate needs to be annealed.
For wood, as suggested above, use a backer block, so the drill doesn't break through when the tip gets close to the surface. Drill wood at a high speed, and clear the chips often.