I think that's your PATS transceiver. It reads the key code and sends it to the PATS module, and allows the engine to start. If the engine will start, its OK.
The key warning chime switch is inside the ignition switch itself. I don't think you can buy it separately, so you will have to replace the entire switch assembly.
Is your trouble that the chime won't function with the key in the switch, and the door open?