LED switch on/off issue with wiring (transistors and grounding)
You should tie the grounds together as you showed in the second image, but you need to move the LED to between the positive terminal of the battery and the collector of the transistor.  Otherwise the transistor can't turn on.

EDIT: As Passerby noted and I missed, you also need a resistor between the Arduino and the base.

But given the LED is 100W, even if we assume it gets a full 9V (I'm assuming the LED in the schematic represents an array or driver module, etc.) it would be over 10A.  That transistor you selected is only rated for 3A.  You should select a different main power transistor, and probably should use another transistor stage between the Arduino and that output transistor to not overload the Arduino pin, as shown below.







simulate this circuit &ndash; Schematic created using CircuitLab