Especially after the battle of White Mountain, what brought Denmark into the war? The battle, from what I've read, seemed to have more or less ended the original Bohemian rebellion that started after the Defenestration of Prague. The leaders of the rebellion were taken care of and roman catholic rule had been reasserted in Bohemia. So why did Denmark feel the need to take part in a conflict over 300 miles from its borders, even if the leaders were both protestant? Hadn't the rebellion already "ended" at that point? (Also, I'm sure I have errors or misconceptions in my question, so feel free to point out any that you see.)
So Christian of Denmark ruled over probably the most powerful Protestant Kingdom at this point of the War (1621). He took a dim view of the Winter King, saying to Frederick:
"Who advised you to drive out kings, and seize kingdoms"!
Despite this Denmark still got involved. A few reasons were behind the decision. First off Christian of Denmark feared that the actions of Frederick of the Palatine had jeopardized the position of Lutherans in the Holy Roman Empire. Christian wasn't alone in this, the ruler of Saxony, Joahnn Georg, got involved for the same reason. They both wanted to protect the position of Lutherans. Both feared that the Holy Roman Empire was being secretly influenced by powerful Jesuits (a key feature in Protestant propaganda) and that the Holy Roman Emperor would attack the rights of Lutherans. Christian of Denmark wanted to be seen as the protector of Lutherans.
There was also practical reasons, for example Christian used his invasion to scare the Hansa (trader states on the Baltic and North Sea) into giving Denmark favourable trading agreements.
This is a very good question, and you should read through some of past posts that answer parts of this question.
Here's one by /u/Bromao/ that answers why Denmark switched sides during the 30YW.
Here's a more strategic view of how the 30YW grew to be a very large conflict, that may better address why it continued beyond the Bohemian revolt.