Here's my suggestion, fairly cheap, quick and efficient.
Pick up a scrap piece of melamine board like here, and clamp it:
Apply a bead of polishing paste (here we have the POL. Wenol, Autosol or Tormek polishing pastes would do about the same):
Spread the bead on the board with a piece of tissue:
The residue of the first single push (Stanley 9½ Sweetheart blade, about one hundred years old carbon steel yet still quite hard at 63 HRc)
Seven pushes, already pretty polished:
18 pushes, rather overpolished (razor edge appeared already a while ago):
Here are three different steels polished, all very razor edges. High carbon steel (the Stanley Sweetheart), low alloyed high carbon steel (the chisel) and M42 HSS (the marking knife):
In the last picture, the melamine surface is already quite clogged. It's no wonder, I started polishing the blades right after a medium grit diamond stone (to compare the result for M42 at HRc 65).
Please consider, you can use any very hard surface (hard epoxy, phenolic, urea-formaldehyde etc ) for this kind of polishing. On those hard surfaces, you can either push or pull the blade. You can also use let's say 2...6mm thick polyurethane elastomer sheet glued on rigid substrate to make a strop, on which you only pull the blade.
Those aforementioned "greasy" amine soap binded polishing mediums work the best (POL, Wenol, Autosol, Tormek) because they will remain viscous and will be spread over the surface under the honing pressure. Water based binders (some commercial mediums are like skin lotion mixed with ceramic powder) seem to abruptly clog and will not really spread evenly over the surface, because they just dry up too soon and become too thick.
For those who have access to raw materials, I recommend Almatis (ex-Alcoa) A 16 SG alumina powder, where SG denotes "superground", where the material is ground close to monocrystalline level. With that you can apply the same kind of binder as with those commercial polishing pastes. It will then be even finer, yet very fast and powerful paste for steel edge polishing.