The ionic conductivity of a coordination compound in an aqueous solution is directly proportional to the total number of ions it furnishes upon dissociation . Let us examine the dissociation of the given complexes:

1. $K_4[Fe(CN)_6] \rightarrow 4K^+ + [Fe(CN)_6]^{4-}$ (Total 5 ions)
2. $[Co(NH_3)_6]Cl_3 \rightarrow [Co(NH_3)_6]^{3+} + 3Cl^-$ (Total 4 ions)
3. $[Cu(NH_3)_4]Cl_2 \rightarrow [Cu(NH_3)_4]^{2+} + 2Cl^-$ (Total 3 ions)
4. $[Ni(CO)_4]$ is a neutral molecular complex and does not ionize in water (0 ions). Since $K_4[Fe(CN)_6]$ produces the maximum number of ions in an aqueous solution, it will exhibit the highest ionic conductivity.