Why won't Craftsman 16.5HP riding mower start?

I have a 1999-ish Craftsman riding mower (16.5HP / 42" cut). My wife started it last week and mowed two rounds before running out of gas. She put fresh gas in the mower, but it wouldn't start. We thought she may have left the key in the "lights" position, so we charged the battery for several hours. That didn't work, so I replaced the battery. It still won't start -- or even "turn over".
I have checked the wiring to the best of my novice ability. The 15A fuse is good (checked at 40x magnification). The spark plug looks fine. The battery terminals were cleaned. Yet, the mower won't start. The solenoid doesn't even click.

Any ideas on what I should do next?