vanity mirror lights project help needed

[ProDave]

The fuse is there to protect the cable. So if your lights develop a fault and they are protected by a 13A fuse the fault current could be in the region of 21A

With a short circuit the fault current will likely be a LOT more than 21A. (typical Zs for ring final circuit 0.4 hms would give a pssc of about 575A)  the fuse needs to be a low rating so the fuse blows before the cable melts.