US5881 Hall Effect sensor not detecting AC field
There are a few things wrong with your approach right now:


The sensor is not sensitive enough. Consider the Biot-Savart law \$B=\dfrac{\mu_0I}{2\pi r}\$, where I is the current and r is the radius from the center of the wire. For a 120W light bulb at 120V (1 A), and the minimum separation to the center of the wire (optimistically 2 mm), you end up with 0.1mT. Compare that to the trigger threshold of your sensor (25mT nominal), and you would need 250A in order for it to trigger.
The light is off. Without current flowing (even if voltage is present) you will not have any magnetic field to sense.
The switch is sensitive to magnetic fields in only one dimension, and you have picked the wrong one. Again, looking at the datasheet, the field must be going through the front of the package, which would mean that a wire would need to be touching the top of the package for any current-induced magnetic field to actuate the sensor


Ideally I'd like to see a capacitor near the sensor, but that's not your problem. That sensor is really just for detecting magnets as a tachometer or an interlock/switch, not for detecting the presence of AC current.