It turns out that a hot solution of alum (ammonium aluminum sulfate), which is available in the spice rack of your local supermarket, will dissolve the steel drill bit without reacting with the brass.

Ammonia is added to solutions of cobalt, nickel, and copper 2+ cations. First formation of the hydroxide precipitate is seen, then of the ammonia complex. Pour 0.1M solutions of Cu2+, Co2+, and Ni2+ ...