Statement (I): This statement is incorrect. Dimethyl glyoxime forms a square planar complex with nickel, not a six-membered covalent chelate, when treated with NiCl$_2$ solution in the presence of NH$_4$OH. The resulting complex, known as a nickel-dimethylglyoxime complex, is characterized by a square planar structure where the nickel is coordinated by the nitrogen and oxygen atoms of two glyoxime molecules.
Statement (II): This statement is true. Prussian blue (Fe$_4$[Fe(CN)$_6$]$_3$) is a coordination compound containing iron in both +2 and +3 oxidation states. The compound forms due to the combination of Fe$^{3+}$ ions and [Fe(CN)$_6$]$^{4-}$ complex ions, leading to a structure with mixed oxidation states of iron.
Conclusion: Statement I is false, and Statement II is true.
Werner’s coordination theory in 1893 was the first attempt to explain the bonding in coordination complexes. It must be remembered that this theory was put forward before the electron had been discovered by J.J. Thomson in 1897, and before the electronic theory of valency. Werner did not have any of the modern instrumental techniques and all his studies were made using simple experimental techniques. Werner was able to explain the nature of bonding in complexes and he concluded that in complexes, the metal shows two different sorts of valency: primary and secondary. Primary valences are normally ionisable whereas secondary valences are non-ionisable.