How would one determine an element simply by looking at its binding energy?
There are a few clues on how to solve this question: 


you are given binding energies. The energies in question are just enough to liberate electrons from their bound state. 
The equation you've written $$E_{nl} = -\frac{Z_\mathrm{eff}^2R_\ce{H}}{n^2}$$ describes the binding energy. The important part is that it can be assumed to be only a function of n and l, that is, of the subshell to which an electron belongs ($Z_\mathrm{eff}$ is a function of n and l quantum numbers. Unfortunately this is probably a somewhat complicated function but that is not very important just now)  


Finally consider the Aufbau principle, outlined in this libretext:


  In essence, you can think of electrons filling in "n+l " order. If you have an equal "n + l"  value for two orbitals, then electrons will occupy the orbital with the lower "n" value.  


which is followed by a nice illustration (shown below). The number of distinct binding energies equals the number of subshells. The energy for emission from each subshell decreases in the same order in which the subshells are filled. All electrons in a subshell share a common binding energy. Following the Aufbau pattern you can identify the highest-energy occupied (valence) subshell, which corresponds to the smallest binding energy. I count 8 subshells and assign the minimum $E_{nl}$ to the 4p subshell (the electron configuration is $\ce{[Ar] 3d^{10} 4s^2  4p^?}$). This means there are six possible atoms that match the number of binding energies, namely Ga through Kr.

If we cheat and check a table of ionization energies we see that the lowest binding energy (-14 eV) matches that of Kr (other values in the table don't match those in the problem since the table lists the ionization energy for multiple electron emissions whereas the problem lists energies for all possible single electron emissions $\ce{A-&gt;A+ + e-}$, starting from the neutral atom, an emission from one subshell corresponding to one energy). You should note the pattern of decreasing binding energies (magnitudes) starting from the 1s electrons (−14326 eV) up to the valence shell (-14 eV). The following diagram adapted from the libretext quoted above illustrates how this works: