XBox 360, no connection to Live, cannot update, cannot factory reset

I have an XBox 360 with a current Live account. It will not connect to our internet and keeps prompting an Update that cannot download. We have tried to factory reset and it does nothing. We have deleted every app. I've tried everything they suggest
on all the Help pages and we cannot get past the error. When we searched the error specific number, nothing is found. Help?

:)

 

Achievements not unlocking for Backwards Compatibility Games

Factory reset is done, now I cannot sign in to Xbox Live from the 360 emulator. I am online with the Xbox 1, playing multiplayer and watching netflix, connection test shows that everything is working.

When I open 360 games, my gamertag pops up as signed in, but not connected to Live.

 

cant connect to XBOX live

I ran the test

I reset to factory settings

I followed all steps

I can get internet connection

But cannot get to xbox live

WHY NOT????!!!

 

Try to apply the offline update:
http://support.xbox.com/en-US/xbox-360/console/system-updates-info

 

We tried that, and it got us further on the update screen than we have gotten previously. It still did not complete and gave us an error.