I have a 1992 Honda Accord DX and I frequently leave my head lights on, or my keys in the ignition because the chime doesn't go off. Does anybody know how to fix this?

Do you have any other chime related symptoms? Like the door ajar indicator staying lit, and/or the seatbelt alarm not functioning properly?
These all share the same alarm module (Honda calls this the Integrated Control Module), and many share a common gound: G401. I would inspect the ground connection but you're probably going to need a that module replace due to an intermittent open internally.

The only chime that works is when I first turn the key... before it turns over.