楽しかったんだ is indeed 楽しい + た (past) + の (explanatory-no) + だ (copula). The のだ part can be translated into English as "It's that ~", "The thing is ~", "I mean ~", "You know, ~", "I tell you, ~" or whatever, but this part itself is in the present tense because it's about the current mood of the speaker. The past marker (た) is attached to 楽しい, but technically speaking, it's "wrapped" in a subordinate clause.
So, is this sentence in the past tense or present tense? It depends on what aspect of the sentence interests you. If you're interested in pure syntax, the main predicate of this sentence, (ん)だ, is undoubtedly in the present tense. However, if you consider the main focus of the sentence is 楽しかった ("it was fun"), and view のだ as an added nuance to convey the speaker's mood, then it would be perfectly accurate to describe the sentence as one about the past. The fun experience definitely happened in the past, after all.
If you ask English speakers whether the sentence "I tell you, it was fun!" is in the present or the past tense, you would likely get a variety of responses, wouldn't you? Some might say the sentence is in the past tense because "I tell you" is just a set phrase that expresses a mood, like "hey". Your question is the same.
If you move the past marker (た) to the end of the sentence, you'll get 楽しいんだった, which is 楽しい + の (explanatory-no) + だ (copula) + た (modal-ta). This is still grammatical but means something you might not expect: Conjugated word + んだ vs nonconjugated word + conjugated んだ