There was a similar situation when I moved into this house, the old boy had wired up a neon across the fan power terminals for the downstairs loo so he could see when the fan was left on.
It has since been removed.
An LED version will be the same but it must be a 230v AC one, lead length may affect how it's installed as you have to be able to remove the cover etc.
Perhaps one that is not flat, a round head that, if you can mount it in the fan chassis and simply drill a hole in the fan front for it to poke through may be a better solution.
However, that fan looks old and may well fail before too long anyway, requiring a new fan and starting again.