When Hezekiah heard the false dilemma offered by Sennacherib's agents, "he tore his clothes and put on sackcloth and went into the temple of the LORD (2 Kings 19:1 NIV)."
The first thing he did was soberly recognize the dire situation that he faced. So many times, I have seen people refuse to recognize the way their lives are spinning out of control, heading for destruction. And so much like the drunk driver oblivious to the fact that he just crashed through the bridge out sign with the pedal to the metal, the stereo blasting and a smile on his face, they continue merrily on their way.
That reminds me of something I once heard Howard Hendricks say: "I've heard people criticize Jeremiah for being the weeping prophet. Folks, at least Jeremiah realized he should weep." Like Jeremiah, Hezekiah realized he needed to mourn, so he tore his clothes and put on sackcloth."
The next thing he did was enter the temple and seek the LORD. He sent a group of men to visit the prophet Isaiah with this description of the awful situation he faced: "This day is a day of distress and rebuke and disgrace, as when children come to the point of birth and there is no strength to deliver them (vs. 3)."
He continued, " It may be that the LORD your God will hear all the words of the field commander, whom his master, the king of Assyria, has sent to ridicule the living God, and that he will rebuke him for the words the LORD your God has heard (vs. 4)."
Finally, he concludes his message to Isaiah like this, "Therefore pray for the remnant that still survives."
In short: He recognized the severity of the situation, he hoped that God would hear how God was being ridiculed (apparently with the hope that God would respond accordingly), and asked Isaiah to pray for the remnant that remained. That's how he responded to the false dilemma. Not a lot of attention on himself.