Ireland had been at least nominally in the possession of the monarchs of England ever since Henry II landed in 1170 and claimed the island by authority of the papal grant of the island by the bull Laudabiliter thereby displacing the previous line of native High Kings of Ireland. One has to say 'nominally', as the extent to which the English actually exercised authority over Ireland varied considerably over the centuries and practically speaking much of Ireland remained outside their direct control unless there happened to be an English army in the field.
Nevertheless since that time the Kings of England had claimed dominion over Ireland and styled themselves Dominus Hibernie, Lord of Ireland. They continued to do so until Henry VIII simply proclaimed himself King of Ireland on the 23rd January 1542, claiming that he always had been King of Ireland and that it was simply that he and is predecessors had neglected to use the title. In truth what had fundamentally changed was that Henry had rejected the authority of the Pope and established his own Church of England of which he was the Supreme Head in Earth. Having thus rejected the Pope, Henry had no wish to continue to be 'Lord of Ireland' by virtue of a four hundred year old papal bull and thus asserted that he was "justly and rightfully King of our realm of Ireland, and ought to have the title, style, and name thereof by right of inheritance".
Thus Henry VIII now adopted as his formal title that of;
Henry VIII, by the grace of God King of England, France, and Ireland, Defender of the Faith, and of the Church of England and also Ireland in Earth the Supreme Head.
although, as is often seen in modern legislation, he specified a transitional period until the "last day of April" during which period the use his former style, without reference to the kingship of Ireland remained valid. The Crown of Ireland Act 1542 was passed soon afterwards confirming the details of the proclamation, although curiously there was later a rival for the title as in 1555 Pope Paul IV issued a bull granting the title to Philip II of Spain.
The title of King or Queen of Ireland as appropriate, continued to be used by the monarchs of England, and later those of Great Britain, until the year 1801 when Ireland was united with Great Britain and became part of the United Kingdom of Great Britain and Ireland. This United Kingdom became disunited in 1922 when Ireland, with the exception of the six counties of Northern Ireland, became the independent state of Eire.
Technically speaking as the independent state of southern Ireland achieved dominion status, the monarchs of the United Kingdom (now the United Kingdom of Great Britain and Northern Ireland) were also the Kings of Ireland or Eire if you prefer. This persisted until 1949 when Ireland became a republic and replaced the monarch with a president as its head of state. A state of affairs later recognised by the Royal Titles Act 1953.
The Proclamation of Henry VIII as King of Ireland
WHERE we be justly and rightfully King of our realm of Ireland, and ought to have the title, style, and name thereof by right of inheritance, and the non-use thereof in our style hath caused much disobedience, rebellion, dissension and sedition in our said realm, to the great impoverishing and peril of destruction of the same, if we had not for the redress thereof put to our kingly hand, as we have done, in such wise, as by reason thereof our said realm (thanks be to God) is now brought and reduced to better order, peace, and civility than it hath been many years past;
And forasmuch as our loving subjects of our said realm, both the prelates, nobles, and commons, do think and determine, that the good estate, peace, and tranquillity, wherein our said realm now standeth, shall the better and longer continue, if we would as we ought of right, accept and take upon us the title and name of King of the same; which to do all our said subjects, of our said realm, by their mutual assents, by authority of parliament holden within the same, have agreed and assented unto, and most instantly desired us, that the said title and name of King of Ireland, together with our said whole realm, should be united and annexed to our imperial crown of our realm of England:
To which their desires and humble requests, for the better conservation of the good peace of our said realm, we have assented, and have caused for that purpose our style to be altered and reformed, as well in the Latin as in the English tongue, as hereafter followeth: Henricus octavus dei gratia Anglie, Franciae and Hibernie rex, fidei defensor, and in terra ecclesiae Anglicanae and Hibernicae supremum caput. Henry VIII, by the grace of God King of England, France, and Ireland, Defender of the Faith, and of the Church of England and also Ireland in Earth the Supreme Head.
And to the intent that our said subjects should not be ignorant of the alteration of our said style, in form as is aforesaid, we have caused this present proclamation to be made, and by the same will and command all and singular our officers, justices and ministers, and all other our subjects and residents within this our realm of England, and elsewhere within any our dominions, that they shall accept, take, and use our style, in form above written, in like form, as they used and accepted our old style before this alteration.
Nevertheless, to the intent that no discord, variance, occasion, trouble, impeachment, or molestation should be had or made to any our justices, officers, ministers, and other our subjects or residents, before they may have convenient knowledge of the change and alteration of our late style; we are therefore pleased and contented, that none of our said justices, officers, ministers, subjects, or other residents within our realm of England, the dominion of Wales, Calais, and of the Isles of Jersey and Guernsey, for omitting of our said title and name of King of Ireland in writs, patents, process, or other writings, to be passed under any our seals, or for nonacceptation or misacceptation thereof, or for any offense touching the same, done or committed, or to be done or committed, before the last day of April next coming, shall be vexed, troubled, impeached, or by any wise molested or troubled, but that all writs, patents, process, or other writings that be passed or shall pass under any our seals, before the said last day of April, wherein shall happen our said title and name of King of Ireland to be omitted, shall be taken, construed, accepted, and admitted to be of the same force, strength, quality, and condition in all things, as they were before the said title and name of King of Ireland was annexed to our style.
And that the non-acceptation or mis-acceptation of our said title and name of King of Ireland, or any acts or things done, or that shall chance to be done, before the said last day of April, by any our subjects or residents, touching or concerning our said title or name of King of Ireland, shall be construed and expounded any offense or occasion of trouble to any of our said subjects or residents; anything contained in this proclamation, or anything that shall be expressed in the same, or any other thing or things to the contrary thereof in any wise notwithstanding.
Proclamation adding 'King of Ireland' to the royal style (Westminster, 23 January 1542) from Paul H. Hughes and James L. Larkin, Tudor Royal Proclamations (Yale University Press, 1964-69)