If I understand you correctly, you want to know how Enlightenment ideas in the Magna Carta and the Mayflower Compact influenced the United States Constitution. You also mention the Bill of Rights, which definitely contains Enlightenment ideas.
The Magna Carta and the Mayflower Compact
The original Magna Carta was drafted by King John in 1215, and it guaranteed the rights and liberties of freeborn Englishmen. In the late 1600s, John Locke described these rights as "inalienable" rights, and you see these words in the Declaration of Independence:
We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.
To John Locke, government was necessary in order to promote the public good. You can see this Enlightenment idea in the Mayflower Compact:
Covenant and combine ourselves together into a civil body politick, for our better ordering and preservation and furtherance of the ends aforesaid; and by virtue hereof to enacte, constitute, and frame such just and equall laws, ordinances, acts, constitutions, and offices, from time to time, as shall be thought most meete and convenient for the generall good of the Colonie
Essentially, the Mayflower Compact provided for a system of autonomous government, and this idea of government ("of the people, by the people, and for the people" as described by Abraham Lincoln in his Gettysburg Address) greatly inspired how the United States Constitution was crafted.
The Bill of Rights and the United States Constitution
The Bill of Rights is the first ten amendments to the Constitution. You may be familiar with a few, especially the first one, which guarantees freedom of speech, freedom of the press, the right to peaceably assemble in gatherings, and the right to petition the government. The first amendment also establishes freedom of religion with a caveat: the government cannot set up an official religion (or show a preference for a specific religion). At the same time, it cannot restrict religious faith of any sort.
The second amendment (which has generated much controversy) is the right to bear arms. The third amendment ensures that you and I do not have to "quarter" or provide shelter to soldiers during peacetime or war (except as prescribed by law). As we go through the ten amendments, we notice a pattern: the amendments provide for the rights of the people living in America. This is a key Enlightenment idea.
The Constitution was greatly impacted by the ideas of three Enlightenment philosophers. These philosophers were John Locke, Voltaire, and Charles Montesquieu. Locke believed in representative government (that is what we have in America, a representative democracy). Meanwhile, Montesquieu advocated for a system of checks and balances, where the judiciary, executive, and legislative branches provided a check on each other. Montesquieu's separation of powers theory became the foundation of the United States Constitution.
Last, but not least, Voltaire championed freedom of speech and religion (the rights you see enshrined in the first amendment to the Constitution). You may have heard that Voltaire spoke these famous words: "I disapprove of what you say, but I will defend to the death your right to say it." In actuality, the words were first written and popularized by Evelyn Beatrice Hall (or her pseudonym S. G. Tallentyre) in 1906. Hall merely wanted to describe the Voltairean values she so cherished.