It's strange that you can't find it, because the first page of results that Google gives me for
magic graph include this set of exercises. Your question is exercise 1.3a.
$K_n$ is magic for $n=2$ and $n\ge 5$.
(Note: this corrects the reference).
Leaving aside the small cases,
the proof for $n\ge 5$ is by the following construction. Since the graph is regular, we may consider arbitrary weights (not necessarily positive) and later add a suitably large constant to all of the edges.
I'm going to make a tweak to the proof.
Enumerate the cycles of length 4 that are contained in our graph, assigning them numbers from $1$ to $N$ (where $N$ is the total number of these cycles). Let the $k$th such cycle contribute $\pm 3^k$ (alternating) to its edges.
Let us check that this set of weights is magic. The total weight of the edges adjacent to every vertex is $0$, because the contribution of each cycle to this sum is $0$, so it suffices for the weights to be distinct. But for any pair of edges, there exists at least one cycle of length $4$ which includes only one of them. Therefore the sums of weights determined by the cycles are not equal, because the weights obtained by our construction may be regarded as $N$-digit balanced ternary numbers and so two edges can only have the same weight if they share all their $4$-cycles.