Your question suggests you haven't understood the importance of the nature of the L2 point.
Useful links to read more, if the brief description below isn't enough:
The L2 Lagrange point isn't an object. It’s a point in space where forces balance out to create a point that James Webb Space Telescope (JWST) will easily stay near. The "halo orbit" means it won't orbit exactly in an ellipse as usual, but in a slightly different varying curve, that needs correction now and then.
And that's the answer to your question.
.....at a sufficient radius around the Lagrange point that it is in perpetual sunlight. That allows it to have predictable solar power....
There isn't such a place as you visualise, there. L2 isn’t an object. Nowhere is "in the shadow", and you can't get into shadow by hugging it closer, because there's no solid object actually at L2, for JWST to orbit, like it would a planet or moon, and be in the shade of.
(Technically, L2 is one of five places where an object will broadly stay in the same position relative to Earth, with very minimal course correction being needed. Some matter may gather at or near L2, but no solid object in the sense you're thinking, able to provide shade and attract an orbiting satellite)
100 metres from L2 or half a million km from L2, it’s all the same thing - perpetual sunlight exposure (or more exactly it’s sliiiightly in Earth's shadow, so somewhat reduced sunlight) - but nothing actually there for it to hide behind.