I have put this together to the stage shown - mirror stuck to MDF with LED strips down each long side of the MDF and wired into the driver - my question is whether there is any need for any sort of earthing of the mirror and if so how that is achieved? thanks

No need for earthing but you will need an enclosed terminal box for the mains connection not just a pair of naked wago's.

 

I would also use an enclosed junction for the DC connections.

 

Personally, I'd get rid of the DC connection outside the driver and connect both DC supplies directly to the driver unit. I think there's plenty of space to do this in the Tagras. That way you've got less connections and LED strips can be a bit sensitive sometimes.