To place Venice in the context of Italy there is A Short History of Italy, edited by H. Hearder and Daniel Waley. It is indeed short (not quite 250 pages). The Roman period is dealt with at breakneck speed in the first two dozen pages, while the Middle Ages and Renaissance are given a bit more detailed treatment to round out the first half of the book. The second half slows considerably as it describes the establishment of the nation of Italy in the 19th century and its history through World War II and the recovery afterwards.
Venice's role in the Renaissance, especially the cultural aspects, can be found in Jacob Burchhardt's old but still respected Civilization of the Renaissance in Italy. For the general reader, Will Durant's The Renaissance, the fifth volume of his monumental Story of Civilization, has the same level of drama and pageantry that is found in Norwich's work.
For those who wish to dig deeply into what shaped the history of Venice, these four books offer detailed analysis of specific aspects: