Hi there, are you getting some kind of error code or message that can help us narrow the issue? As I can see you're a silver account, have you updated and verified security proofs before? If not, you may need to do so by following the steps here:
I would suggest using the mobile phone as it's the fastest and most convenient, after you have done that you may need to delete the profile from the console (profile only), clear the system cache, restart the console and re-download the profile.
So when doing the new password did you happen to select the two-step verification method? This will mean you've created an 'app' password which will only be used to sign-in to the Xbox 360 console, see here for more information: