Indeed, we have a “Constitution”. It is called “The Torah”. It contains all necessary laws and teachings of ethical behavior. Unlike the US Constitution, which was written by “Founding Fathers”, the Torah was given to us by our Divine Father in Heaven.
Regarding government branches, let’s begin with the “Executive Branch”. We are taught that ideally the Almighty would be the “One Executive” who not only gave us the Torah, but also would continue to lead us and teach us the correct way to live.
The mention of our being led by a king was not meant to be the ideal. Only if we would not be on a high enough spiritual level to follow God’s leadership — as relayed to us by Moshe, Joshua and Prophets that followed — we would be allowed to appoint an “earthly leader”, called a king. In fact, this is what happened historically, and we were ruled over by many kings throughout history, such as Saul, David, Solomon and many others. From the time of the Babylonian Exile when the First Temple in Jerusalem was destroyed, there have been no more kings that lead us, but rather this leadership role has been filled by the most scholarly and revered Rabbis in each generation. They are generally known as the “Gedolei HaDor”, the “great ones of the generation”.
Regarding the “Legislative Branch”, we need to be clear that no new Torah laws may be added to the one, perfect Torah. Any additional laws that could be made were considered “Rabbinical” in nature, since they were instituted by a Jewish Rabbinical Court, called a “Beit Din”, which means a “House of Judgment.” Most of these Rabbinical laws have been enacted throughout history for the purpose of helping ensure that we adhere to the Torah. Some Rabbinical laws are to commemorate specific historical events, such as Purim and Chanukah, or to enhance our experience of the Torah laws, such as lighting candles for Shabbat to ensure that we are not eating in the dark.
Regarding the “Judicial Branch”, these same Jewish religious courts (Batei Din) hear all court cases and render legal decisions. So it is interesting to note that the function of the Beit Din is “dual purpose” — both to enact certain new regulations, and to adjudicate legal cases and issues.
Of course, much more information is necessary in order to understand the exact functioning of each of these branches throughout history and in our times, but I hope this brief reply provides some helpful insights into an authentic Jewish governing system.