It sounds like you will have to only plug in using a wired internet.
If you want to spend a some money to get it fix you can learn the steps to request a repair at support.xbox.com/.../repair-process. If you try to get it replaced or have someone else replace it other than Xbox that will terminate any warranty on your Xbox.
(LFJ)Xbox Ambassador - Question? Feel free to ask!2012 Fall Beta Dashboard Tester
If you have a Xbox 360 Slim (which as built-in wifi) and you are asked to attach an external networking adapter or an ethernet cable, then the internal wifi has malfunctioned and your Xbox 360 Slim will need to be repaired.
mine has done the same only since the beta update