Let's say, we have a loop of wire of length \$L>0\$. Let's say, hypothetically, it is a loop of non-ideal wire. That means that the wire has a resistance per unit length \$ρ>0\$. Let's further assume that there is a changing magnetic flux through that loop of wire. Faraday's Law of Induction, as it is widely understood, dictates that a current will be induced in that loop of wire. Let's call that current \$I\$ (where \$I>0\$).
Now, for the sake of argument, let's divide that wire into infinitely many segments of infinitesimal length. Let each of those segments have length \$dl\$. That means that each segment will have resistance \$ρdl\$, and since each segment has the same current \$I\$ passing through it (KCL), the voltage drop across each segment should be \$Iρdl\$ (Ohm's Law).
Starting at any point in the loop of wire, we can sum up the voltage drop across each infinitesimal segment for the entire loop (the segments are in series):
\$\int\limits_{0}^{L}Iρdl=IρL>0V\$
Now, since we start at any point in the loop and come back around to the same point, that means that the voltage at that point (relative to itself) is both \$0V\$ (trivially) and \$IρL\$, which is a contradiction since we assumed that neither of \$I\$, \$ρ\$, and \$L\$ is \$0\$.
This is essentially a long way of saying that KVL finds itself contradicted. But I took this long way to avoid answers such as "KVL simply doesn't work with magnetic fluxes". But how can it not work? What did I do wrong in all those steps? Which assumption was incorrect?