I was having this issue for quite some time, & have found a solution. I wanted to share this solution for any that may be experiencing the same problem now or in the future.. (Getting the "Can't Connect to Your DHCP Server" message. Not being able to connect to Xbox Live using an Ethernet Cable).
Firstly go to Test Network > Advanced Settings > Mac Address > Clear ( it should ask you to restart your console. Wait for the restart. After a couple minutes unplug the Ethernet cable and plug your modem directly into your PC/laptop & open the command prompt (windows > search > CMD ). Type into the prompt "ipconfig/all" take a picture with your phone of the the category " Ethernet Adapter Ethernet" (you'll need to note the IPv4, DHCP, &DNS). (Keep Your Xbox On for now)
-Go back to your Xbox, plug in your modem to the Xbox directly through the Ethernet cable. Go to test network > Advanced settings > IP Address > Manual. Enter all the info from the notes / picture you took. (IPv4, DHCP, DNS) just use the same DNS number for the Secondary DNS number if there's only one.
-Test Network, (it should take a little longer) & now say: "Can't connect through Static IP" (as opposed to "Can't Connect to Your DHCP Server").
Test again (not sure if necessary, although I tried three or four times). Now turn off your Xbox..
-Turn off your Xbox (wait a few seconds & unplug the power from the back of the Xbox). Disconnect the Ethernet from the Xbox while off. Unplug your Modem. After you restart your modem, during its reboot restart your Xbox. When your Modem has connected to your network go ahead and plug it back into your Xbox through the Ethernet cable..
-Now go back to test Network, try again & it should say the same thing: "Can't connect to Static IP". Now go back to Advanced Settings > IP Address >& click Automatic (Wait a couple of seconds, clicking Automatic should now resort to what you entered previously manually.) Go back & test network, it should say " Everything is good ".
(A few things to note: I knew it was not the modem, for I just purchased a new modem & had switched internet providers & still encountered this problem. Even with the new hardware & new services. I thought it could have been my Xbox's internal Wifi apadter, since one day everything was fine & it just stopped working / connecting through Ethernet while I was using it. But then Connecting through Wifi still worked.)
Hope this helps.