Could anyone please tell me what to do to find Aryth?

each room in aryth's spire has a) an assigned colour (you have to pay attention to the text), and b) a left and right mirror

Each mirror in each room leads to one other room and always that room (eg, Sapphire room's left mirror always goes to Bronze room.
More than one room's mirror can lead to the same room (eg, Silver Left and Jade Right both lead to Amethyst room)

You can pretty easily make a map
"okay, I start in X room, I go Left, record room colour, repeat. If I end up back in one of the rooms I already passed through, I know where the left mirror goes, so I go Right in that room, etc."
What you're trying to do is find the correct path through the rooms to reach Aryth's room at the end