How does Potassium Hexfluoronickelate(IV) exist in a stable state?
To quote from the comments: you can get things into an amazing variety of oxidation state given the right environment.
In the case of maximal oxidation states, particularly above $+2$ or $+3$ in the $3d$ transition series, the right environment (at least in simple species) involves complexation with ligands that are highly electronegative, capable of pulling electrons even from a strongly positive central atom, and resistant to oxidation themselves in an electron-poor setting.  Meaning, in practice, oxide or fluoride ligands.
We can therefore roughly classify high oxidation state complexes into oxo and fluoro complexes.  Oxo complexes are more likely to be seen with early transition metals, where the central metal ion contributes few $d$ electrons and there are a lot of vacant orbitals with which the strongly $\pi$-donating oxygen can form back-bonds.  The best known examples being chromate/dichromate ($\ce{Cr(VI)}$), manganate and permanganate ($\ce{Mn(VI), Mn(VII)}$), and ferrate (most commonly, $\ce{Fe(VI)}$).  Even cobalt reaches its apparent maximum $+5$ oxidation state with an oxide-ligand species, $\ce{[CoO4]^{3-}}$.
Later transition metals, which have lower maximum oxidation states due to increased electronegativity/ionization energy, have more $d$ electrons coming from the cationic shells and thus are no longer so stabilized by $\pi$ back-donation.  Then the fluoro complexes, with a relatively $\pi$-inactive ligand, take over.  In the $3d$ series cobalt, cited above as forming an oxide complex with the formula $\ce{[CoO4]^{3-}}$, also forms a +4 fluoride complex $\ce{[CoF6]^{2-}}$(1).  Fluoride complexes are also seen with both nickel and copper in oxidation states above $+2$, including the $\ce{[NiF6]^{2-}}$ ion mentioned in this question and the copper(III) species $\ce{[CuF6]^{3-}}$.  Zinc in the $3d$ series is limited to oxidation states of $+2$ or less, but if mercury has a tetrafluoride $\ce{HgF4}$ (disputed), it's no coincidence that this too is a fluoride.
Reference

J.W. Quail and G.A. Rivett, &quot;Fluoride Complexes of Tetravalent
Cobalt&quot;,  Can J Chem 50:2447 (1972).