On the Dell support site, you should be able to enter the service tag code, found on the bottom of the laptop. Once you punch that in, it will give you the "as built" spec of the laptop, including a full list of hardware and software specific to that exact one.
If the NIC is supposedly installed correctly, check the device manager to see if it's listed as working or if there's a problem. Red X or Yellow ! means disabled, or troubles. If it says that it's fine, then it's not likely a driver error. You can also just do the usual "uninstall" from the device manager, and reboot to let it redetect the nic and auto-install it.
Check in the bios to make sure the onboard NIC isnt disabled. Also, are you running on battery when you are testing it? Some Inspirons that I've worked on disable the onboard NIC when running on batteries to conserve power. Check your power settings on the NIC also in the device manager, and make sure it's not set to turn off the device to save power.. try changing your power scheme also to either presentation mode or "always on".
Do you get a physical link light when you plug in the ethernet cable? A light is a good sign, no light means that the physical connection isnt establishing.. can be one of two things. Either the jack is damaged, or the cable is damaged/wrong. The NIC on your desktop PC could have an autosense port on it, meaning that it will accept a crossover or a regular patch cable.. if the cable IS a crossover, then the laptop may not work.
Bring up a command prompt, and try to run "ipconfig /all". If you get an error about the IP configuration, then your TCP settings are corrupt. If not, then they are fine, and it's something else causing the problem.
Umm.. lemme think some more about this.. I love a good challenge.