Ah, but you see, it would also be important to specify which medieval Kingdom of Italy you are asking about.

The first Kingdom of Italy, as it emerged in the fifth century, occupied a strange existential space somewhere between a restructured part of the Roman Empire and an autonomous client-state. It ended when Emperor Justinian decided to forcibly re-integrate Italy into his Empire; infighting among the increasingly autonomous Romano-Gothic ruling class had offered him a chance to reassume direct control over the peninsula.

The second Kingdom of Italy was extinguished in a similar fashion by Charlemagne. Now, the latest literature that I am aware of (up to and including including researched published by Massimo Montanari at the University of Bologna in 2015) can only agree that there no real consensus as to how or why the Lombards were able to enter Italy so easily. However, enter they did, and were rapidly integrated into what was left of the old Roman senatorial class. The Lombards would contest Italy with the Byzantines from the 6th to the 8th centuries, while also being extremely politically unstable; the peninsula was divided amongst a number (anything between five and thirty-five) power-brokers called Dux, a latin word meaning military leaders (generally rendered in english as "Dukes") who came into conflict with each other in every conceivable way. Thus the "Lombard" state of Italy was generally characterized by a bizarre cyclicality: after the death of a strong sovereign the Dukes begin fighting against each other, the Byzantine military governors in Latinum and Romagna make gains against the Lombard state, the Dukes saw the need to stand together, and eventually rally behind a new King. The cycle then pretty much repeats at the death of the King.

Why was Italy so politically unstable? I would answer, "why not?" Recall Italy's Roman heritage: it is not a consequence of happenstance that the center of Europe's greatest empire is a peninsula characterized by dense urbanization, high agricultural productivity, and strong internal and external trade links. This the two post-Roman kingdoms of Italy an enormous potential for internal instability: with relatively small power bases, aristocrats could easily challenge the state. The state, such as it existed, even in times of tranquility could never adequately affirm itself.

Given this instability, it is somewhat predictable that it took the Lombards a century and a half to consecrate control over Italy. King Desiderius (d. 786) got very close to finally kicking the Byzantines out of Italy: a successful offensive saw the encirclement and neutralization of the Byzantine army in the Pentapolis (that to this day is called "Romagna") thus allowing Desiderius to march south against Rome unopposed. Fearing a loss of independence for the bishop of Rome, who had been pretty much guaranteed a free hand to rule the "Duchy of Rome" in exchange for Byzantine overlordship, Pope Adrian excommunicated Desiderius and asked Charlemagne to come to his aid. Charlemagne, even though busy conquering what would later become Germany, was flattered and immediately marched for Italy.

Charlemagne's conquest of the Kingdom of Italy is another interesting story, but the Kingdom he left behind wasn't necessarily any more stable than the one he found. Much like the rest of post-Carolingian Europe, Carolingian Italy too was divided and run by a group of "Margraves" whose office quickly became dynastic after the Emperor's death. The title of King of Italy rapidly became contested in a convoluted back-and-forth between the Guideschi, a dynastic party from the rugged highlands of Central Italy, the Aleramici, hailing from the hummocky western foothills of the Alps, and the Unruochings, from the northeastern border march of Friuli.

However, what had been the heartland of Lombard Italy prior to Charlemagne's conquest never really bought into the carolingian system. Characterized by a complex hydrography consisting a series of parallel waterways flowing down the Alps across a broad plain to the river Po, "Lombardy" was a densely populated and urbanized region. Curiously, it had been largely absent from the near-endemic civil strife that had catarterized post-Carolingian Italy.

Lombardy too had been assigned a Margrave whose became dynastic. However, this dynasty, the Obertenghi, were surprisingly passive. It's important to understand that the densely populated and urbanized heartlands of the Po basin sustained a stable (some would argue growing) population living in cities, where only with difficulty authority could be concentrated in the hands of a single person. Royal representatives and margraves alike had to contend with clergymen, judges, and lawmakers seated in the cities who were elected, in the Roman fashion, by colleges of prominent citizens. Indeed, thousands of large and medium sized landowners split their time between country estates and urban residences, while small farmers, aided by the shallow waterways crossing the plain and a weak legal precedent for obligations to their landlord, regularly traveled to town in order to bring their goods to market and purchase finished goods made by urban artisans. All this contributed to the liveliness and bustle of urban life. This led to a paradoxical situation where if we consider there was anarchy in the border marches where single strongmen reigned supreme, in the the urban heartland where no one person was unquestionably in charge there was law, order, and some degree of prosperity.

Much like when happened during Charlemagne's conquest, with such a fragmented political scene was only a matter of time until a foreign actor was prompted to enter the fray. Adelaide and Lothair, Queen and King of Burgundy, had been in conflict with Berengar of Ivrea for the title of King of Italy. However, after Lothair's death, Berengar imprisoned Adelaide, who in response asked Kaiser Otto, the ruler of Germany, to intervene.

Otto descended into Italy and defeated Berengar, married Adalaide, and then took two looks at the political system before coming to the conclusion that the Italian urban system of government was the best system of government. Otto tasked the urban clergy with the administration of Italy, partially mirroring what he had instituted in Germany; where he ruled by traveling around the country with a retinue of scribes and administrators who would link up with local clergymen in order to govern. A number of advantageous new charters and privileges were also issued to the urban bishops, importantly with regards to fixing the tenancy of church land to a single lifetime; although intended to endear the Ottonians to the churchmen by forcing renters to renegotiate agreements with their ecclesiastic landlords each generation, this also allowed church land to be granted as a reward for administrative or military service, with the unintended consequence of entrenching the urban aristocracy who saw these renegotiations as a matter of mere protocol before doing away with them altogether.

The end result was an enormous amount of autonomy being amassed by Italian urban bishops and the institutions they headed. Arnulf of Milan, a high official of the archdiocese, wrote a "Modern History" of Lombardy's principal city in the second half of the eleventh century. Arnulf described a complex society, complete with a "middle class" of rambunctious young landowners called iuvenis, as well as a class of rustici, perhaps comparable to yeomen of the British isles. After the extremely Italo-phile Emperors of the Ottonian dynasty ended their reign and were succeeded by the Salians, the Italian clergy instead affirmed that they no longer recognized the Imperial succession. They didn't even propose a candidate they thought was more legitimate, they just said "Nope!"

I wrote about the development of early Italian Communes in this answer you might be interested (you can skip over the first section though, I only re-iterate what's been said here.