electric circuits - Can Kirchhoff's Voltage Law be applied here?

First, consider a simple circuit comprising of an ideal battery of EMF: E and a single resistor of resistance R.

Imagine that the switch is opened in the circuit. In the battery, some internal mechanism drives (by applying a force) negative charges to one terminal, leaving positive charge on other terminal, at equilibrium an opposing electric field is generated inside battery. No further accumulation of charges takes place.

As soon as the switch is closed, electrons get another 'path' to go from terminal having higher potential to that having lower relative potential through the resistor. An electric field (conservative in nature) is set up across resistor.

Now, the work done by conservative fields (present inside the battery and across the resistor also) on any charge over a closed (or cyclic loop) path is ZERO. That's why I think Kirchhoff's Voltage Law works.

My problem is that why do we apply Kirchhoff's Law for Inductors as well?

In an inductor, an EMF is induced by changing magnetic flux. The EMF is simply the work done by the NON CONSERVATIVE electric field,(produced by changing magnetic flux in the inductor coil) on moving across the a particular path i.e across the inductor coil.

Since this field is non-conservative, according to my analogy given at the starting of the question, this work done across closed path on the charge shouldn't be zero. But in books I see that Kirchhoff's Law is applied here! How can this happen?

Any help will be appreciated.