Once nobody came to their support unlike the First Slesvig War, the Danish must have realised that its hopeless, as Prussia alone could, if it wanted to, number an army ten times their numbers. Not to mention that their armies were of course of higher standard and quality.
So why then did the Danish go through with their plan anyway? Why fight a hopeless war? Even if they thought that some ally might still come to their rescue, the risk seems not worth taking.
The short answer is that it was a gross miscalculation bordering on stupidity by the ruling National Liberal government, headed by Ditlev Gothard Monrad.
To set the stage, when Denmark changed the Constitution in 1863, there was an understanding that it could lead to war, especially after Bismarck issued his ultimatum. The hope of winning the war against the very superior Prussian army largely rested on the hope of controlling the seas, stalling the enemy, and give the great powers of Britain and France a chance to intervene in favor of the Danes.
This failed for a variety of reasons. First of all, Danish war preparations had not been made soundly with this strategy. Rather it was infected with nationalistic fervor that was against giving up land, and it wanted to hopelessly hold the border at the southern edge of Slesvig. This plan clearly failed, and let to the initial rout.
Now the more interesting part of your question is, why did the Danes continue - specifically, why did they not accept the peace suggested by Bismarck at the London conference in May, in which Bismarck offered a fairly lenient peace, that would place the border about where it is today.
This is where it gets somewhat embarrassing. The problem is that the government, including Prime Minister Monrad, who had clear mental issues and broke down under the high pressure situations, could nota gree what it wanted. The delegates travelled to London with conflicting instructions from the king, who wanted to return to pre-war status quo before the NOvember Constitution, and the government, who didn't want to accept the partition of Slesvig. The result was a clusterfuck of a peace conference, in which the Danes turned down Bismarck's peace proposal without offering a clear alternative, to the immense consternation of the British peacemakers. Meaning they had to suffer yet another series of defeats before they finally surrendered, and accepted whatever peace Bismarck offered.
So in short, it was political stupidity and poor leadership. Denmark was brought in a hopeless position by politicians who did not know what exactly they wanted, and therefore they failed to formulate a meaningful policy.