The only reason why I bought Xbox for another month was to play Black ops 2 but yet it does...

Another Call of Duty game or another copy of Black Ops 2? Just because you can play, let's say, Black Ops 1 without a HDD does not mean that the developers didn't make it a requirement for Black Ops 2. Can you play single-player or is this a multi-player
only issue? Again, you're not stating what happens when you try to play the game.

i bought xbox live gold for 1 month.. but can't play call of dudy black ops on xbox live.. why???.. its says.......The current profile is not allowed to play on xbox live

It's when I play black ops 2 it does it I don't know why... Thanks

If you have 2 factor authentication enabled 360 doesn't know what to do with that

You need to use your app password

https://support.xbox.com/en-US/help/xbox-360/security/set-up-app-password

Call of Duty: Black Ops 2. Yes it's Call of Duty, but it does offer zombies as well as multi and single player. I also Haven't played Call of Duty Multiplayer since Modern Warfare 2. I've got the original Black Ops, but I bought it for $2 and only play Nazi
Zombies on it. I like to play Call of Duty, I just don't buy every release like some people do.