Once Irish monasteries started growing to thousands of inhabitants, (laypeople and monks), and became important centres of learning, commerce and administration in the 700s Irish kings began to build their households there instead of on isolated fortified settlements.

This is actually not quite true. While there is an example of a King-Bishop during this time period (Cormac mac Cuilennáin was simultaneously a bishop and King of Munster in the first decade of the 10th century), Irish kings did not make any attempts to consolidate or centralize their power until at least the 12th century. While some monastic settlements could be described as urban centers, Ireland was a thoroughly rural society comprised of relatively self-sufficient farmsteads dotted across the countryside; the Irish never even established villages until the 13-14th centuries. The disparate pattern of settlement and economic structure of Irish society was the base on which its political superstructure took form. Accordingly, Irish hierarchies of kingship were incredibly decentralized and appear to have been relatively mobile when compared to the courts of European monarchs in the late Middle Ages; an Irish king and his retinue of noble clients were often moving between the homesteads of other clients, as the provision of hospitality was a very important aspect of the Irish political-social system. Lacking any state apparatuses to extract taxes or rents, Irish lords rented cattle to their dependents in exchange for hospitality (among other things), essentially sustaining the warrior-aristocracy with food and labour on a contractual rather than institutional basis.

An Irish king's power was dispersed on a geographical level as well. They did not draw their power from a unified, central settlement but from three major sites scattered around their tuath: the hereditary homestead of the king's lineage, the ecclesiastical site that their family patronized and the inaugural mound on which kings assumed their office in a public ritual. So yes, Irish kings in the early medieval period did rule in part from their farmstead, but because of the decentralized nature of Irish society and the ritualistic and almost priestly role of the early Irish king, they weren't really doing that much "ruling" in a modern context. To give you a quick rundown, Irish kings appear to have originated as priestly figures whose duty was to uphold the metaphysical wellbeing of their kingdom by the abidance of strict taboos & prerogatives, possessing an unblemished body and making good judgements in legal cases. Later on, starting around the 8th century, they begin to transform into regional warlords who are more concerned with subjugating their neighbouring kingdoms. This coincides with the Viking Age in Ireland, which led to a further militarization of Irish society and the further expansion of kingly power with the infusion of Scandinavian allies, weapons and tactics.

So we've established that Irish kings did, in fact, "rule" from their homesteads, but what would they have actually looked like? As far as I'm aware, there isn't archaeological evidence that kings' households were any more lavish than a typical nobleman's. They may have lived either within a farmstead or a rath (ringforts that had been built during the Iron Age) or crannog (a fortified artificial island) but the actual house of an Irish king doesn't appear to have been palatial or grandiose. The only description of a king's house in a historical source that I found researching for this post only describes the personal items of the king, rather than the building itself:

And he was in his sleeping-place then, and his riches were all there with him, as it was customary for the kings to have cubicles of yew about them, that is, a partitioned place, for their bars and cases of silver and their cups and goblets to give service at night, and their brandub and fidchell games and their bronze hurley-sticks to use by day. Feradach had many treasures, and he loved them greatly; but he had acquired them by evil means, for he would not hear of much or little gold or silver, in the possession of either powerful or wretched in Osraige, without confiscating it to take away that wealth, to ornament those treasures.

Sources:

• “Fragmentary Annals of Ireland, FA 4”, trans. Joan Newlon Radner. Last edited 2008-09-05, http://www.ucc.ie/celt/published/T100017/

• D. Blair Gibson, “Chiefdoms, confederacies, and statehood in Early Ireland” in Celtic Chiefdom, Celtic State, ed. Bettina Arnold & D. Blair Gibson (Cambridge: Cambridge University Press, 1995)

• F. J. Byrne, Irish Kings and High Kings. (Dublin: Four Courts Press, 2001)

• Thomas R Kerr, Finbar McCormick & Aidan O'Sullivan, The Early Medieval Archaeology Project (EMAP): Project Report 2013 - The Economy of Early Medieval Ireland. (December, 2013)