Yes
I think you need a meter to test the red pair.
Turn power off
Set meter so is buzzes when probes touched together.
Connect them up to the red pair.
Pull switch cord
Does buzzed start/stop when cord pulled?
If yes it's the switch wire and its been joined.
Thanks.
However, the mirror light is fine, the only thing I can't get working (or even try yet) is the ceiling light. Should I just do the same as you said but test by flicking the wall switch and not the pull cord?