Hi cara d.,
If the LAN controller is enabled in BIOS and not showing up in Device Manager (neither under Network Adapters nor Other Devices), it indicates towards a hardware failure and the system board will need replacement.
However, to rule out any software corruption, I would suggest you to take the computer back to factory condition or re-install the OS (if comfortable).
Note : All data will be lost in the process. Please backup important data before proceeding further.
You may refer to link Here for performing a PC restore and taking the system back to factory condition.
If you are unable to do a PC restore, you may try re-installing the OS manually using the Dell OS re-installation DVD. You may refer to article Here for step by step instructions for re-installing Windows. Once the OS is installed, you install the drivers in correct order as mentioned Here and check the LAN connectivity.
If the issue persists, you may consider replacing the motherboard as Ethernet port is located on system board. However, if your system is out of warranty, a cheaper alternative would be to use a PCI LAN card or USB LAN adapter. Both of them are available at nominal prices.
For installing PCI card, you may refer to Service Manual Here.