These mirrors usually need a permeant live and a neutral and they then have switch on the mirror itself.
If there was access to a loft above then should be able to make a suitable junction up there to get the wiring required at the mirror.
I suppose the cable could then be brought down in trunking assuming any external influences have been taken into account, not exactly going to look great but each to their own.
However not sure how you would make for provision for entry of the cable into the mirror as all the ones I have seen are suitable for rear entry only...
This work is notifiable as it is an extension to circuit in a special location. Whatever work is done it needs to be tested, and fully compliant with BS7671 (Is type A RCD Protection in place?) and then notified to local authority.
Really it's the kind of work that ideally undertaken during a refurb of the whole bathroom