I think they tried. Schleswig was mostly Danish although there was a significant German population in the south. Holstein was mostly German and more valuable to Prussia. However Bismarck wanted more room around the proposed Kiel Canal that would allow a more direct marine connection with the Baltic Sea from the west.The Danes should really have tried to negotiate a division of the territory along the language boundary at the London conference, but they seem to have been incapable of coherent action.
Palmerston bowed to public pressure and declared that Denmark would "not be alone" if attacked by Prussia and Austria.The Prince of Wales was recently married to the very popular Princess Alexandria of Denmark. Ironically the Queen, as well as much of the Cabinet, was opposed to any British intervention and Palmerston folded telling his supporters "The Queen wouldn't let me." She told the angry Princess that her country was England now. The Royal Navy wasn't keen either citing ice on the surrounding seas.
Queen Victoria and Her Ministers in the Schleswig-Holstein Crisis 1863-1864 on JSTOR