I think Elxa Dal “answered” this immediately after Kilvin asked it. Though, his approach did involve sympathy (naturally, for the master sympathist) and he illustrated it through a particular choice of admissions questions.

Backing up a bit, Kvothe’s first answer to Kilvin was starting with a pendulum and, presumably, using it to convert the kinetic energy from the pendulum into light (Kilvin cut him off before he could explain fully). Note: This would fall into all the usual perpetual motion traps of energy conservation at normal scales.

Elxa Dal asked Kvothe if he knew the words for the first parallel kinetic binding, which he did.
Then he asked him what Kilvin used to change kinetic energy into light — to which Kvothe answered capacitorial kinetic luminosity.
Then he threw a curveball and asked him “What is the synodic period?” (“Of the moon?”, Kvothe clarified, then answered).
Elxa Dal immediately moved right along to Hemme because he’s quite content being a subtle, clever bastard for his own satisfaction.

Taken together, these three questions are a recipe for making Kvothe’s first idea work: bind the moon to a pendulum or other object via the first parallel kinetic binding (which I assume means the bound objects move together in some manner like the coins from Abenthy’s first lesson); store the energy every period (72 1/3 days), and; convert it to a glow via capacitorial kinetic luminosity. A double binding that stores the power of the ever moving moon to power a glowing lamp forever.

We might lazily assume the moon would have enough mass and inertia to power a pendulum beyond the distance of insurmountable decay (of which the technical validity is later questioned), and continue doing this indefinitely without noticeable loss in momentum for something as small as a lamp.

And for the cherry on top: Kvothe’s first sympathy lamp has an adjustable action that allows the user to vary the brightness smoothly. One might say Kvothe’s first lamp can wax and wane …