The adapter light is actually 2 lights. One is internet the other I beleive is usb activity. Both out? Does your internet work on pc? Try replugging both ends in? Switch the usb port on the wii, if your ethernet line is not fully "clicked" into the adapter the lights will be off, also try a different ethernet line, one that you know is good,and also try another port on your router.
Last but not least there are times I can't connect for whatever reason, be it servers are busy, or my wii gets freaky, somehow in my settings it reverts to the old wireless connection. I have to go in and reset it to use wired. This happened quite afew times. Usually after failing to connect wired 3-4 times it does it. Whether its supposed to or not I don't know. Hope this helps.