Naw the JWST won't stay there after it loses power.

> The points L1, L2, and L3 are positions of unstable equilibrium. Any object orbiting at L1, L2, or L3 will tend to fall out of orbit; it is therefore rare to find natural objects there, and spacecraft inhabiting these areas must employ a small but critical amount of station keeping in order to maintain their position.

Nothing will. The Lagrange points are only quasi stable.

L4 and L5 are "stable" -- but of course on timescales of billions of years even Earth's orbit isn't guaranteed to be stable.