Dell's BIOS has a setting in it for audible keyboard click. I own a Latitude D600 with the same setting. When you turn it on, watch for a message that says...
"Press x to enter setup"
Or the like. Press that key, then read CAREFULLY the instructions to navigate around until you find a setting for audible keyboard sound, keyboard click sound, or something that sounds like it could not be anything else.
DO NOT change any other setting. These settings are hardware-level and some are non-reversable. Some will destroy your data. If you CANNOT find a setting that has to do with the key-click sound, JUST TURN IT OFF!
Don't do anything! Just turn it off! Hold down the power button for five seconds and it's off.
I make it sound like it's a giant deal, but I just want to make sure that you won't see options and start changing them. I've had too many friends who go and change BIOS settings and fry their computer because they changed some option for their hard drive or video card.
It's not a difficult thing if you show some intelligence. So far you seem a superbly intelligent person, AnnieHall. But I've had to disable this option before. I bought my Dell second hand and the previous owner had the key-click sound on. I had to disable it manually.
If you want, send me a PM or a message on Yahoo. I'll make myself visible to you and we'll get this annoyance fixed!