Look, this situation would be a national tragedy of the highest order. Bush & Cheney would be dead, but in all likelihood a large number of innocent people would have died as well--probably the first and second ladies, a large number of staffers at the white house, and probably at least one member of the cabinet.
In this kind of terrible incident, I think a nuclear response might be considered by more people than the dismissive voices on this thread. Ultimately, nuclear weapons will be left on the table--but not used. 9-11, even a worse attack where Flight 93 hit the White House with Cheney and Bush inside, or some other variation, would not justify a nuclear reprisial--at least not one immediately. And as calmer heads prevail, the nukes will remain on the table but off the battlefield.
One of the greatest tragedies of 9-11 is how coldly and divisively it was turned into a political advantage instead of a national tragedy. Politics as usual and the whole paradigm of "rule the largest peace" could have been ended right then and there. In this regard, Bush blundered away what could have been the best presidency since the second world war.
I'm lapsing into politics here, sorry. But the point is that President Hasert would understand that the United States would need to respond to the terrorists--but his handling would probably be more competent and with a recognition that he was the president of all Americans. The United States would accept NATO's help and invoke article 5, and NATO would work together to combat international terrorism. Russia, understanding that a great opportunity has come to smash Islamic Militants emerging in its own turf, would probably join in, possibly joining NATO outright as a result.
Bin Laden might still escape into Pakistan; but if nothing else the world will have worked together to respond to Fundamentalism instead of "rule by the largest piece"