Did they have a "successful kingdom" in Italy? I would suggest not.

I am drawing mostly from the New Cambridge Medieval History chapters on the subject in vols. 1 and 2.

As Moorhead points out in Vol. 1, the first mention of the Lombards in Roman sources suggest that their numbers were very small compared to other groups and cemetery evidence seems to bear this out. He goes on to point out that 90 percent of the male graves included arms, suggesting implicitly that the few Lombards (compared to the rest of the population) had to use force of arms to maintain their hegemony. Furthermore, they entered Italy after the harrows of the Gothic Wars and the Justinianic Plague and even then had difficulty securing the port cities and could never contend with the Byzantine navy.

Delogu, who has written a couple of other excellent chapters and books about the subject, points out that royal administration was not nearly as evenly spread. He states:

Nevertheless, King Liutprand's concepts [of royal administration] were more aims to attain (my emphasis) through royal government, than objective reflections of the institutional machinery for the time being. IN reality, political and jurisdictional powers were not as evenly ordered as the iudicaria system might suggest. Not all of the Lombard territories, for instance were equally subject to royal authority. This extended over the greater part of northern Italy, comprising the Po valley, Trentino and Friuli; in that part of Italy only the provinces of the Byzantine Empire—Venetiae, the Exarchate and the Pentapolis—escaped royal control. South of the Apennines, Tuscia, approximately corresponding to modern tuscany, was also subject to the king, perhaps in looser form. Further east and south, the duchies of Spoleto and Benevento, two vast regions named after the seats of Lombard dukes who presided over them, did not fully recognize the authority of the king (again, my emphasis), even though they shared the laws and customs of the kingdom.

All over Italy, local dukes held titular power rather than the king and there were many wars fought after the Lombard invasion.

It was only in the 8th century when kings began to be able to exert more royal control over their kingdom and by that point the much more powerful Frankish kingdom began to become enmeshed in Italian politics.

At the same time, the cities of Italy began to reassert themselves as major centers and became loci of resistance to royal authority. One thing that Delogu doesn't get to here (but I believe he does in one of his books) but people underestimate the number of old Roman families that would have persisted in the region.

TL;DR

The kingdom was not "successful." The Lombards were a small minority and kings had to contend with powerful nobles to assert their authority.