How can i fix this my vanity lights?
OK... what you need to do is:

1) after turning off the power, remove the two screws above and below the middle socket of the vanity lights that are not working. Examine the wires in the junction box for any burn parks are disconnected wires. Redo any connections that look bad.

2) Remove the cover from the other side of the vanity, the side that's working, and examine the connections for the wires from other lights to where they hook up to the white and black wires from the switch. Look for burnt or disconnected wires. Redo any connections that look bad. Put a light bulb in each side and turn on the power and flip the switch and see what happens. Good luck.