This letter can be seen as an echo of the Declaration of Independence because it appeals to the idea of fundamental human rights and asserts the right of the people to oppose a government that does not protect those rights.
In the Declaration, Jefferson says that the purpose of government is to secure the inalienable rights that people have been given by God. Put differently, he is saying that there is a natural law that is above all governments and which they must obey. In his letter, King argues that people should resist a government that creates and enforces unjust laws. He says that
An unjust law is a human law that is not rooted in eternal law and natural law.
In saying this, King is putting forward the idea that a government and the laws it makes are only legitimate if they are in accordance with natural law. This is the same thing that Jefferson is saying in the Declaration.