This is a little outside my comfort zone, but I'll give it a shot. The short answer is "no". Even after 1171, the kings of England rarely had any real influence in Ireland, despite their nominal rule as 'Lord of Ireland'. English settlers and aristocratic families, most notably the various branches of the Geraldine clan, were the conduit of English rule and often acted as governors/lieutenants on behalf of the English monarch, but even at the height of English power in the mid-13th century they only had control over maybe two-thirds of the island. Ireland wouldn't be properly politically united in any meaningful sense until the early modern period, when the last Gaelic Irish earls in Ulster went into voluntary exile in the early 17th century (the so-called 'Flight of the Earls').

So what about pre-Norman Ireland and the 'High Kings'? Well, as you've probably read in other places, pre-Norman Ireland was divided up into tribal kingdoms known as tuatha (sing. tuath), of which there were up to 150 at one point. In many ways its better not to think of these in political terms - the translation in Latin chronicles was not regnum ('kingdom') but plebs (people) and a tuath was probably more important for one's cultural and social identity than one's political one. By the end of the first millennium, several 'over-kingdoms', composed of several dozen tuatha had formed, ruled by kings who held the loyalty of 'sub-kings' of various tuatha which made up their spheres of influence through hostages, tribute-taking and ritual gestures of submission. From c. 800, the most powerful of these kings began to take the title of 'King of Tara', with Tara being a hugely symbolic (but politically worthless) abandoned Iron-age fort in modern Co Meath. Throughout the ninth and tenth centuries, this title, symbolizing the paramount king on the island, passed between several branches of the Ua Néill family, who were based across counties modern Meath, Weatmeath, Tyrone and parts of Cavan, Armagh and Fermanagh (roughly North-Central Ireland). In later histories, particularly the seventeenth-century compilation known as the Annals of the Four Masters, this symbolic paramount kingship was turned into the 'High-Kingship', although in reality it never meant anything like control of domination over the whole island, for warfare and competition between kings was an ever-present in Irish society in the period.

The closest anyone actually came to uniting the whole island in the medieval period was probably Brian Boru (Brian Boruma mac Cennetig). From an obscure royal family in modern Co Clare, Brian built on the achievements of his father and older brother to establish himself as King of Munster (southern Ireland), before launching into a series of expansionist wars, using the wealth and military power of the coastal Hiberno-Norse cities of Limerick and Waterford to overwhelm his rivals. After gaining control of Dublin in 999, following the battle of Glenn Máma, Brian was pretty obviously the most powerful king in the country and the King of Tara, Mael Sechnaill mac Domnaill, submitted to him in 1002. In 1005 he successfully led an army north into Ulster, the only part of Ireland which had not yet submitted to him, received submission from all but one of its major kings and gave his protection to the church of Armagh, the country's prime ecclesiastical institution. It was in this year that his visit was recorded in the famous 'Book of Armagh' as Imperator Scottorum - Emperor of the Irish. The claim isn't strictly true, since the King of Cenel Chonaill, modern Co Donegal (extreme north-west of Ireland) had not submitted, but it nonetheless represented a substantial achievement. Brian returned north in 1006 and collected tribute to be given to Armagh; although Cenel Chonaill did not formally submit, Brian marched through the territory unopposed.

This was the apogee of Brian's power, however, and subsequent years saw increasing opposition and rebellion, most of which Brian manage to subdue, including the rebellion of Leinster and the Dubliners in 1013/14. This rebellion culminated in the famous battle of Clontarf, on Good Friday 1014, when Brian's forces were victorious just outside Dublin, but the elderly King himself (in his mid-70s) was slain.

Brian's 'empire' collapsed with him and no king would come close to the control he had on the country for more than half a millennium. He fundamentally changed politics on the island however, and subsequent kings tried to mirror his successes by declaring themselves rí Érenn, although chroniclers and annalists usually added the words co fressabra - 'King of Ireland with opposition'. But even Brian's rule possessed little stability, and was based fundamentally on an annual circuit with his army to ensure everyone stayed in line. Even then, he only had one or two years (1005/6) when he could be said to have been without opposition. There was simply no institutional structure upon which to build any sort of stable, long-lasting polity on the island.

Further reading:

Seán Duffy, 'Brian Boruma (Brian Boru), c.941 - 1014' in Oxford Dictionary of National Biography is a good outline of Brian's life and campaigns.

Donnacha Ó Corráin, Ireland Before the Normans is very old, but still the best introduction to Irish politics in the period before the English invasion - it's very readable and makes a complex subject very easy to understand.