I would suggest you to try out this fix:
1) Goto System Settings.
2) Select Storage.
3) On your Primary Storage Device Press the 'Y' Button.
4) Select 'Clear System Cache'
5) Wait for it to Clear...
6) Now attempt to Sign in.
7) You should stay signed in.
Here is one More Possible Fix:
1) Goto System Settings and Select 'Storage'.
2) Click the Storage device that Contains your Profile.
3) Goto your profile and press 'A'.
4) Click Delete Profile. It Will then ask 'Delete Profile Only' or 'Profile and Items'.
5) Select 'Delete Profile Only'
6) Once deleted press the Xbox Guide Button(Middle of Controller/ Big Green X)
7) Make sure you are signed out of all profiles and click 'Download Profile'
8) Enter all your Windows Live Credentials when It asks you too.
9) Wait for it to download
10) Once Downloaded you should be able to Sign-in to XBOX LIVE.
try this solution to get the network settigns back to its default state:
1)Turn off your Xbox 360 console.
2)Wireless connection only: Unplug the Xbox 360 Wireless Networking Adapter from the USB port (if applicable).
3)Turn on your console.
4)Press the Guide button on your controller, go to Settings, and then select System Settings.
5)Select Network Settings.
6)Select Wired Network or the name of your wireless network, if you are prompted to do so.
7)Select Configure Network.
8)On the Additional Settings tab, select Restore to Factory Defaults.
If all of those doesnt work i would suggest you to use this link to help you find a solution: