Demolish the damaged tower
I'm with those that say that they'd probably demolish the second tower.
Change the attack order
I think that you're approaching this from the wrong angle. A less successful 9/11 would not, in and of itself, make the response stronger. If you want a stronger response, you should change how it proceeded. Instead of going in the actual order, change it to
- White House
- Pentagon
- World Trade Center North Tower
- World Trade Center South Tower
The first three succeed and the fourth fails (as really happened). So Bush, most of the cabinet, and the top generals are dead. Cheney is President. Cheney panics and lights off nukes.
This may not be realistic. Cheney may not be quite that bad. But it's believable. Changing the responder from Bush to Cheney could change the response in a way that more moderate changes in circumstances could not.
Add a second attack
Another possibility. Make 9/11 less successful and weaken the response. Yes, I know that you want to strengthen the response. We'll get there.
There are a number of ways to make the attack less successful. A quick example would be to change the result of the presidential race. President Gore may have read the Tom Clancy book that inspired the 9/11 attacks and made the single change necessary to prevent them: lock the plane's cabin door and not open it for any reason. Perhaps this only works in three of the four attacks, so the North tower is still destroyed but the other targets are not hit.
With an unsuccessful 9/11, we could expect the same kind of results as the previous failed attack on the World Trade Center in 1993. I.e. pretty much nothing. With a partially successful 9/11 where the only success is a failure in a mostly effective system, it's quite possible that the response would have been smaller. Particularly if we combine it with a different President.
With a less powerful response, it seems quite possible that there would have been another attack. Put it in late 2003 or early 2004. After the failure of the previous response, the President has to respond much more strongly. Or put the response in 2005 with a new President.
Again, this may not be realistic but is believable. It is a possible result of Gore beating Bush in 2000 which could be explained as simply as eliminating the butterfly ballots in Florida. No butterfly ballots and Gore almost certainly wins easily without recounts. Why no butterfly ballots? A Republican was angry about how they caused Dole to lose votes in 1996. Of course it didn't affect the result in 1996, but a suitably anal-retentive personality might not have cared. A small butterfly change could have led to a big change in 2000. Entirely in line with chaos theory.
Why an alternate history?
If you want a strong response in 2040, why do you need people to respond more strongly in 2000? You don't seem to need a change in 2000 to justify the world that you describe in 2040. So why bother? Make things happen in 2016 and later. You have twenty-five years to build the world that you want. Why do you need another fifteen years of changes first?
The kind of changes you're trying to put into 2000 would be more likely to cause a nuclear war in 2004 or 2014 than 2040. If 2051 is the tale you want to tell, then tell that. Because we don't know what's going to happen in 2016-2040, any changes you want to make will seem more realistic.
Civil War
I'm not clear on why nuking Afghanistan would cause a Civil War in the US. A Civil War based on foreign policy seems unlikely. More likely would be an increasing number of domestic terror incidents and the domestic responses to them. You could set that anywhere.
The US is already (in the real world) increasing in polarization. It seems quite possible that the blue states and the red states might split on how to respond to domestic terror incidents. They already have very different opinions on things like the Patriot Act, surveillance, and the meaning of religious tolerance.
Note that parts of what we think of as blue states might stick with the red states. Examples: upstate New York; Pennsylvania outside Philadelphia, inland California and Washington. The more liberal voters are concentrated in urban areas of the US. For certain states, taking the urban areas out of the equation would change the state's lean. New York City and Philadelphia are examples of cities that could switch out of their existing states easily because they fit the adjacent states better. Eastern Washington might fit better with Idaho than Seattle.
This has a historical basis. In the real US Civil War, West Virginia split off from Virginia to join the North.