The words 'Taoiseach' (Prime minister) and 'Dail' (parliament) are two of the most common Irish words routinely used by journalists in Ireland and Britain (somehow). In fact the 'only' irish words routinely used (don't get me started) in the Anglophone press. A literal translation might be 'leader, but prime minister is perfectly fine. The same could be said for Chancellor Merkel - it's a title!
I did a bit of research and prime minister translates to "Príomh-Aire" and is used to refer to prime ministers of other countries, (e.g. An Príomh-Aire na Spáinne nó an Príomh-Aire na Breataine, the Prime Minister of Spain or the Prime Minister of Britain). Also from CSPE in school, An Taoiseach is defined as the head of government or prime minister in the constitution of Ireland, so I guess "Irish Prime Minister" is also a valid translation. The Irish language newspaper Foinse is a contemporary online publication with plenty of examples.
I would still suggest that that is an infelicitous translation in English. If, for instance, I wanted to refer to the monarch of Japan, I would see nothing wrong with simply leaving the word Tenno in Japanese, thus emphasizing his difference from other monarchs. It would also, however, be acceptable to refer to him as the Emperor, emphasizing his similarity with other emperors throughout human history. To refer to him all the time as the Japanese Emperor or the Emperor of Japan, however, would be stilted in English, and I would certainly recommend a student not use it. Similarly, when I refer to the Taoiseach in conversation in the United States, I usually simply refer to the title in Irish, because the Irish do, apparently, make this distinction in the language (not that I see much difference in function). That's fine, and if I have to explain to someone what a taoiseach is, I simply say it is the Irish term for their prime minister. I'm not going to go around saying Irish prime minister, though.
I think if you were writing in English about another country's leader/monarch, you would likely use their language's term for him/her on first reference, with an explanation after it (in brackets) to explain in English what this means, and thereafter you could refer to him/her by either their official name in that language, or in the English version.
For example, and this is an English-language newspaper meant for people outside of Ireland: "Today in Dublin, the Taoiseach (the Irish Prime Minister) announced that he is, in fact, the orange-eating, blue-coat-wearing Pol (Paul) from Duolingo's online course in Irish. The Taoiseach went on to say that he isn't very fond of blue coats, but that he always eats before the crab."
That is, more or less, how we used to write things when I worked in government.
I'm just going to use Taoiseach from now on whenever I see it, since it is going to take me forever to remember how to spell it.
As this is a course about Ireland and the Irish language, writing simply "Prime Minister" should suffice, ratrher than "Irish Prime Minister". Both are accepted answers, but the "Irish" seems to me to be superfluous. We are not talking about comparative political sysrems, when it might be relevant to differentiate between the Irish Prime Minister as compared to the Prime Minister of another country.
I get that this is the translation but I always found it strange to expressly teach ones to translate it to "Irish prime minister" saying as we always use (An) Taoiseach even in english conversation/publications and even the nuacht in English. Feel like they should have covered this in the "notes" for this section, but it seems the developers left a note to themselves about what the section was about, rather than populating it with useful stuff. But fair do's.
Strictly speaking, England doesn't have a Prime Minster of its own - though "the British Prime Minister" is also a bit terminologically inexact!
Cá bhfuil Príomh-Aire Shasana? - "Where is the English Prime Minister?"
Cá bhfuil Príomh-Aire na Breataine? - "Where is the British Prime Minister?"