A candle arm on a mirror has to be set out from the flame backwards, not from the carving forwards.
The cup is the fixed point. Everything else - how far the arm reaches, where it leaves the frame, how high it sits - follows from putting the flame in a position where it stands clear of the frame above it, clear of the glass beside it, and where the drip pan sits under the candle rather than beside it.
So the arm's projection is decided first, and the carving is designed around the arm rather than the arm being fitted into a gap in the carving.
The fixing is a socket. Each arm is turned with a spigot at its root that drops into a lined socket cut into the frame - not glued in, not screwed through the face. That means the arm carries its load into solid timber, it can be lifted out for cleaning, and if you would rather use the mirror without candles the socket is closed with a small carved cap and the frame reads as complete.
The drip pan is turned as part of the arm rather than added, so there is no seam for wax to sit in.