The annexation of Spanish Florida had nothing to do with the war of 1812. We would not gain Florida until 1819. Our primary concerns were the British encouraging the Indians to attack us in the Northwest Territory. They were also impressing our sailors and had been for some time. The struggling economy was also tough. Wars have a tendency to bring us out of depression, most notably was WW2 helping get us out of the Great Depression. The War Hawks in Congress also applied great pressure to President Madison. Madison was more of a philosopher, less a military man. He was constantly pressured to do something by Congress or he was seen as weak and ineffective.
