How to tell which species has the highest ionization energy?
Like most chemical conundrums you need to simplify the problem using some knowledge of chemistry.  

First there are two electron configurations. $\ce{Ne, Na+}$ and $\ce{Mg^{2+}}$ all have the configuration ${1s^2 2s^22p^6}$. $\ce{Ar, K+}$ and $\ce{Ca^{2+}}$ all have the configuration ${1s^2 2s^22p^6 3s^2 3p^6}$. 

For $\ce{Ne, Na+}$ and $\ce{Mg^{2+}}$, $\ce{Mg}$ has the highest atomic number so of these three, it would be most difficult to ionize $\ce{Mg^{2+}}$.

For $\ce{Ar, K+}$ and $\ce{Ca^{2+}}$, $\ce{Ca}$ has the highest atomic number so of these three, it would be most difficult to ionize $\ce{Ca^{2+}}$.

Now it is just necessary to compare the ionization of $\ce{Mg^{2+}}$ to $\ce{Ca^{2+}}$. Since the $2p$ orbitals are closer to the nucleus than the $3p$ orbitals, it will be harder to ionize $\ce{Mg^{2+}}$ than $\ce{Ca^{2+}}$.