Is Kirchoff's Voltage law valid to derive current for LC circuits?
It appears to me that they are in fact using Kirchoff's Voltage Law, using the differential relation between current and voltage for an inductor with self-inductance, L:

$$V=-L\frac{\mathrm{d}I}{\mathrm{d}t}$$

Note that this is in fact derived from Faraday's Law in this case, cf. the line:

$$\oint \vec{E}\cdot\mathrm{d}\vec{\ell}=-\frac{\mathrm{d}\Phi}{\mathrm{d}t}=-L\frac{\mathrm{d}I}{\mathrm{d}t}$$

Where the LHS of the equation is the closed loop voltage and the RHS is the familiar differential relation from the center of the equation.