Follow up: The buzzing noise may not be coming from the ignition switch. It may be the solenoid. Again, if I jump start it, it will start.

Get the battery charged & Load Tested.
You didn't provide any model # of your mower, but many of these only have a 3 AMP charge system. That means it'd take about 10 hours to charge a fully run down battery.

What likely is happening-
The battery has enough charge to activate the solenoid-
As soon as it tries to crank the starter motor, it pulls the battery voltage so low, the solenoid drops out-
Battery voltage then increases enough to activate the solenoid-
Repeat......
Kind of a machine gun effect.

This post was edited by bill_kapaun on Sun, May 4, 14 at 16:26

Thank you. I'll try that. That does sound like the most plausible explanation.

Ugh. Battery is good and fully charged. What you mentioned above fits the description. Any thoughts why jumping works? It seems to be some other electrical related issue.

Thank you! Cleaning up the connectors did the trick

The same principle that Bill explained was doing it. The continuity afforded by the loose or dirty connection points simply could not carry enough "juice" to operate the solenoid AND power the starter all at the same time.