So I've now made a list of ruling authorities of Ireland. It's probably not ideal, since I'v'nt checked wether every one of these pre-Republic ruler, got coins minted (in their name) and it's only as completer as to the earliest coins on Numista. There is also a few problems with some of the high kings, since I couldn't figure out from when they reigned, only when they died. But hope it's a help at least :D
Ruling authority of Ireland
High Kings of Ireland
Mael Sechnaill mac Domnaill, 979–1002; 1014–1022
Brian Boru, 1002–1014
Donnchad mac Briain, died 1064
Diarmait mac Mail na mBo, died 1072
Toirdelbach Ua Briain, died 1086
Muirchertach Ua Briain, died 1119
Domnall Ua Lochlainn, died 1121
Toirdelbach Ua Conchobair, died 1156
Muirchertach Mac Lochlainn, died 1166
Ruaidrí Ua Conchobair, died 1198
Lords of Ireland
John (1177–1216)
Henry III (1216–1272)
Edward I (1272–1307)
Edward II (1307–1327)
Edward III (1327–1377)
Henry IV (1399–1413)
Henry V (1413–1422)
Henry VI (1422–1461)
Edward IV (1461–1470)
Henry VI (1470–1471)
Edward IV (1471–1483)
Edward V (1483)
Richard III (1483–1485)
Henry VII (1485–1509)
Henry VIII (1509–1542)
Kings of Ireland
Henry VIII (1542–1547);
Edward VI (1547–1553)
Lady Jane Grey (1553; disputed)
Mary I (1553–1558)
* Philip jure uxoris (1554–1558); his role as King of Ireland was reinforced by the Treason Act 1554
Elizabeth I (1558–1603)
James I (1603–1625)
Charles I (1625–1649)
Charles II (1660–1685)
James II (1685–1689)
William III (1689–1702) and Mary II (1689–1694)
Anne (1702–1714)
George I (1714–1727)
George II (1727–1760)
George III (1760–1800)
Act of Union
George III (1801–1820)
George IV (1820–1830)
William IV (1830–1837)
Victoria (1837–1901)
Edward VII (1901–1910)
George V (1910–1922)
Irish Free State:
George V (1922–1936)
Edward VIII (1936)
George VI (1936–1949) Deminished
Sources: https://en.wikipedia.org/wiki/Monarchy_of_Ireland';
- Jamtrup