Dan correct Most likely to the A terminal or something similar on the switch. You can take a volt meter or even a test light and connect between the two solenoid wires and turn the key on and see if you have voltage or the light comes on. If not then move the black/negative wire to a good ground on the engine/frame and see if it gets voltage on the meter/light then.
Does it? Perhaps the voltage goes directly to a bad connection on the wiring harness and stops there. You need to "walk the circuit" with a voltmeter until you identify any weaknesses or defects. Guessing and supposing is no good!
Why are you using a 9 volt presumably transistor type battery instead of the battery on the equipment? That may not have the capacity to energize the solenoid.
I'd measure the voltage at the solenoid to ground and see what happens when you turn on the ignition switch. I'd use the battery on the equipment and test from point to point to identify any weaknesses or defects.
Then measure the voltage from the other side of the solenoid to ground. When you turn on the ignition, that should be zero volts or thereabouts. If you have significant voltage with the ignition on, you have a poor ground connection.
Seattle, I agree about using the equipment battery for testing, but the 9 volt battery test is mentioned in some service manuals, Briggs is one of them.