Inspiron 5100 anatomy

That's correct.  The i5100 should have two pre-installed antennas.

That's what I thought, but that also means I'm running out of possibilities on why I can't get signal to the laptop using the internal wifi card. I installed the card so I didn't have to have the other adapter sticking out the side of the PCMCIA slot and risk busting it off (I'd find a way to do it sooner or later). I made sure the card was properly seated, the leads are on correctly, the proper and up-to-date drivers were installed (even tried the older ones when that didn't work), and made sure of proper configuration of the wifi card, yet it won't detect any wireless networks (it will briefly connect occasionally and it will rapidly fall off). It is recognized and installed without a hitch, but won't detect any of the wireless networks. The external card picks up the networks without a problem. Any suggestions?

Have you run the diagnostics in the Dell Wireless Utility?  It might be a problem with the card.  You might also try reseating the card as it might not be fully seated for some reason.

They just snap on, kind of like the snap on a pair of jeans.  Just be careful not to break the antenna wire itself when you disconnect them from the card.   Personally I get a fingernail under them and gently pry them up.  To reconnect them just line up the connector on the wire with the spot on the card and then press down gently but firmly until they snap into place.

Already have done that, did them both again just in case, still no change. Diagnostics show the card is working perfectly and the card is seated properly. I'm starting to wonder about the antennae themselves.