Whilst you might achieve a hard surface sufficient to cut timber, the life span of the cutting edge even if you could sharpen it would be extremely short lived. Remember this surface is only a few hundreths of a millimetre thick. It would require you to create the cutting edge first, then case harden, then try to polish without removing too much of the hardened material. Additionally you could not sharpen without rehardening.

Your best bet would be to use SAE 4340 grade steel, annealed, quenched then drawn at 400 degrees F (204 Deg C). This will provide a brinell hardness number of 580.

Alternatively use SAE 1300 grade steel, annealed, quenched then drawn at 400 degrees F (204 Deg C). This will provide a brinell hardness number of 480.

In both cases the temps need to be accurate as an increase by 300 deg F will result in a 100 to 80 Brinell loss respectively.

(Ref: Mark's Standard Handbook for Mech Eng 8th Ed)