First, a slight suggestion - you have potentially grossly understated the casualties possible. If the strike is time to occur at noon during a presidential inauguration (for maximum literary effect, just as the president finishes his oath and shakes the Chief Justice's hand), you would kill the entire crowd. While the Trump inauguration was (relatively) sparsely attended, and only totalled on the order of 200,000, the first Obama inauguration pulled somewhere near 1.8 million.
Of course, while that is the best/worst case (depending on which side you're on) it also occurs during maximum security, so it's arguably the hardest time to pull it off successfully. On the gripping hand, it's anyone's guess if the security measures in place for an inauguration are geared up to preventing a nuke. There's lots of room for creativity here.
With that said, such an attack would have the immediate effect of drawing the country together on an unprecedented scale, and would make the post-Pearl Harbor and post-911 responses look like languid tea parties. As mentioned, the Designated Survivor policy would be in place, so at least one legitimate member in the chain of succession would be available. As a result, there would be little likelihood of a military assumption of power, either official or unofficial.
While it's true that the 3 branches would be pretty well decapitated, the actual damage to most of the country is minimal. Communications, for instance would be largely untouched, and the military would respond as it always has, with the Pentagon destruction providing slots to be filled almost instantly. And the loss of the Pentagon would only really affect the very top of the chain of command (the Joint Chiefs, especially) but most of the operational commanders (command, corps, divisional and below) would survive. I suspect most state governments would take a hit, since I expect most governors would be in Washington. And, since we're assuming a very popular president-elect (large crowds, remember?) you should presume most of Congress would be gone. While this would make passing laws to respond to the attack a bit difficult for several months, I suspect that temporary Executive Orders, plus laws on the books allowing emergency/wartime responses would be interpreted in favor of doing what needs to be done.
So, on to the question of the survival of the nation.
Well, it's pretty clear that the nation would survive (in the short run) just fine. One answer suggested that the inability to respond successfully to a mass casualty incident such as this would delegitimize the government by showing it to be inadequate. While I might be wrong, I suggest that this is just silly. While horrified, the rest of the nation would, I submit, recognize that there is pretty much no way to mount a successful response. There would be any number of individual errors which could be criticized post-mortem, but that sort of scale is simply beyond what any emergency plan can handle without required a monumental, sustained effort which would be grotesquely expensive and wasteful until it was needed.
What would happen, I think, is that the primacy of the Federal government would be greatly enhanced. This is a trend which has been in place since the needs of WWII followed by the Cold War, with the expansion of Federally-empowered rights/entitlements which have characterized the last 50 or so years. The country would effectively shift to a war footing, but without an easily-identified external enemy. Internal surveillance, already boosted enormously post-911, would really kick into high gear. Defense/security spending would rise dramatically, with most of the increase aimed at internal security. After all, we already have a WFS (Whole Freaking Shitload) of nuclear warheads, but we don't know who to send them to - unless some group is dumb enough to credibly claim credit for the attack. Shortly before they get hammered.
In many respects, these changes can be seen as an acceleration of trends already in place. Given this view the ultimate effect of such an attack would be to hasten the end of the nation, simply by speeding up the "natural" forces which should be assumed to one day drive us into decline.