Kirchhoff's law in variable regime is not an easy subject, and the lectures on the subject are not always clear. One approach that seemed consistent to me is that of Anuparm Garg in "Classical electromagnetism in a nutshell" (Princeton)
(This is not the only possible approach)
He considers that the lumped circuit is formed of black boxes with an input $A$ and an output $B$ such that the circulation of the electric field on a path outside the black box does not depend on the path followed. This assumes that the magnetic field is negligible outside the box. He then defines the voltage $V\left(t\right)$ as the circulation from $A$ to $B$ on a path outside the box:
$V\left(t\right)=\int_{A\ Exterior\ path}^{B}{\vec{E}\vec{dl}}$ on an exterior path.
For an inductor formed from a perfect conductor, the electric field is zero within the conductor. If we define a closed path which goes from $A$ to $B$ through the conductor and returns from $B$ to $A$ through the exterior, Faraday's law tells us that $\int_{A\ in\ the\ inductor}^{B}{\vec{E}\vec{dl}}+\int_{B\ Exterior\ path}^{A}{\vec{E}\vec{dl}}=-\frac{d\emptyset}{dt}$
The first integral is zero (zero electric field) and the second is $-V\left(t\right)$
Therefore $V(t)=\frac{d\emptyset}{dt}=L\frac{di}{dt}$
He also find $V (t) = Ri$ for a resistor and $V (t) = Q / C$ for a capacitor.
To find Kirchhoff's law, it suffices to write that the circulation of the electric field on a closed path outside the black boxes is zero.
Sorry for my poor english. It is not my native language.
