As with most of the other respondents, I am going to say a firm No.
Let's walk through the entire scenario. You have a high level adviser with a team in place in one of the most secure areas in the United States, presumably with access to the President, Vice President, and all of their staff, but the President is not part of this conspiracy and will not cooperate to launch nuclear weapons.
Your first hurdle is getting the authorization codes you need. It is presumable that you could replace the "Football Carrier" with one of your people, but the football is only one part of the recipe to launch Armageddon. To authorize the attacks, the President also has the "Biscuit," a gold credit-card sized piece of plastic that must be physically broken to reveal a code used to authorize the attack. Without the President on board, you couldn't authorize a launch. The President's identity is also verified by SecDef, so unless SecDef is in your conspiracy, you have that hurdle as well.
Let's sidestep that issue. Say you incapacitate the president, and have the VP on your side, using his football and his "biscuit" to authorize the strike, or managing to give the real President sufficient evidence that he believes a strike is needed. You send out the nuclear launch command to NORAD. Various nuclear facilities are activated, verified, etc, and prepped for the launch. Given that we have never gone to the brink with nuclear weapons since WW2, it is possible that operators will refuse to launch. Just from sheer probability, some will fail to launch for various reasons. At least a few weapons will launch under any scenario where the President (Or VP with the President out of the way) authorizes a strike.
However, it would probably be easier (with the military connections and access you have) to secure a "Boomer" sub. Russia is almost guaranteed to counter-strike if they are attacked, and one the nukes are in the air, the US will have no options left but to strike back or die quietly.