If your wife's pc connects to the outisde world, then you certainly already have a NAT DHCP server. From now on, I will assume you are under Linux, since you call the Ethernet IF eth0.
If you wish to diagnose your problems, you may try a manual connection via these commands:
sudo service network-manager stop
sudo ifconfig eth0 up
sudo dhclient -v eth0
At this point, you should start seeing dhcp offers rolling in.
Normally ethernet connections are quite sturdy, and they do not require tweaking of parameters to work. But if you get no DHCP offers, such tweaking is called for. In this case, you should identify your ethernet card by means of the following command:
lspci | grep Ethernet
which will tell vendor and model (make sure you write down the all-important code inside the square parentheses, [xxxx:yyyy]. Now check if you have a proper driver for this card:
sudo lshw -C network
(if you do not have the program lshw, install it by means of: sudo apt-get install lshw, or sudo pacman -S lshw, or sudo yum lshw...). If it says UNCLAIMED in the Ethernet part of the output (careful, there will be a part on your wifi connection), it means your pc does not have a suitable driver for your ethernet card. This can happen because your card is too new, too old, or requires some kind of proprietary software which your distro feels you have to ask for explicitly. If it does not say UNCLAIMED, check your driver's name.
Now you may search Google with your distro name, your card code [....], your driver's name, and the keyword PROBLEM.