If the eject button is broken and doesn't eject the disc tray out, from the Xbox 360 console, then you can replace the eject button, or eject the disc tray without using it. The eject button on my Xbox 360 console also stopped working, so as a temporary fix, i ejected the disc using the Xbox's dashboard menu.
1)Power on your Xbox 360 console, with a controller connected to it.
2)Once the Xbox 360 dashboard is on, go to the disc icon, if the game disc is inside the disc drive, press X on your controller, to eject the disc. If the disc tray is empty, simply press A on the controller to open and close the disc tray. This way makes having a eject button pointless because you can eject the disc tray, without having to use the eject button.