The problem is well known:
When the leds light current flows through the ground wire.
As the wire resistor can not be zero there is a small voltage drop which modifies the measure (LM35 = 10mV per Celsius degree)
The solution is simple and is only hardware :
Use a ground wire for the LM35 and another one for the leds.
The two wire being connected in star mode only on the arduino board.