How would I calculate the Normality of a 0.00167 molar solution of KIO₃?
Hmm.. I think that your data is wrong, I get that the normality:molarity ratio must be 5 in this case.

This ratio (I call it the $\eta$-factor) for a compound participating in a reaction is the net change of its oxidation state.

In this case, one iodine in $\ce{IO3-}$ (+5) becomes one iodine in $\ce{I2}$ (0). The net change in oxidation number is 5, so $\rm normality=5\times molarity$. Normality is like the reducing/oxidising "power" of a solution--or the concentration of electrons that it can furbish/absorb.

Your density, your $\mathrm{g~L^{-1}}$, or your value for normality is wrong here.

Note that one needs to be careful with this reaction. It is comproportionation, where the same element is oxidising itself (Basically, two forms of the element in different oxidation states are reacting with each other). We can have some trouble when dealing with the products of such a reaction, though dealing with the reactants is fairly simple, as shown above. I'm writing a detailed self-answered post on this later.