According to Genesis 10:10, the city of Babel was originally built by Nimrod, one of the descendents of Noah's son Ham.  

The beginning of his kingdom was Babel, Erech, Accad, and Calneh, in the land of Shinar. (Gen 10:10)

The Ishtar Gate, and the connected processional way, was one of the architectural improvements that Nebuchadnezzar II made during his 42 year reign in Babylon.

This smaller reconstruction of the Ishtar Gate is located in Iraq near the original location of the city of Babylon.  It was built by Saddam Hussein in 1987, along with a restored palace and ancient city wall replicas, as part of a museum dedicated to bringing public awareness back about the glory of ancient Babylon, and fashion himself as a modern day Nebuchadnezzar.

There is also another life-sized reconstruction of the Ishtar gate at the Pergamon Museum in Berlin, completed in 1930 based on the archaeological findings of Robert Koldewey.  A detailed description of his archaeological expedition and findings of the whole city of Babylon, can be found in his book "Excavations at Babylon", available from Amazon books:  https://www.amazon.com/Excavations-At-Babylon-Robert-Koldewey/dp/1298501571/

Inscription on the Ishtar Gate.

Nebuchadnezzar, King of Babylon, the pious prince appointed by the will of Marduk, the highest priestly prince, beloved of Nabu, of prudent deliberation, who has learnt to embrace wisdom, who fathomed Their (Marduk and Nabu) godly being and pays reverence to their Majesty, the untiring Governor, who always has at heart the care of the cult of Esagila and Ezida and is constantly concerned with the well being of Babylon and Borsippa, the wise, the humble, the caretaker of Esagila and Ezida, the first born son of Nabopolassar, the King of Babylon, am I.

Both gate entrances of the (city walls) Imgur-Ellil and Nemetti-Ellil following the filling of the street from Babylon had become increasingly lower. (Therefore,) I pulled down these gates and laid their foundations at the water table with asphalt and bricks and had them made of bricks with blue stone on which wonderful bulls and dragons were depicted. I covered their roofs by laying majestic cedars lengthwise over them. I fixed doors of cedar wood adorned with bronze at all the gate openings. I placed wild bulls and ferocious dragons in the gateways and thus adorned them with luxurious splendor so that Mankind might gaze on them in wonder.