Translation:Finally he remembered that today is his wife's birthday.
That's an excellent question. Semantically it's obvious that nifal should be used here, but when I thought about it in more depth I couldn't really put my finger on why. The best explanation I can deliver at this stage is that the action of recalling something spontaneously (as in something popped up) is using nifal, while paal (זכר, זוכר) should be used when he had and retained this knowledge throughout. You can definitely say הוא זכר שהיום יום ההולדת של אשתו, but using סוף סוף indicates there is some action which happened immediately after this adverb, meaning, his recollection of his wife's birthday. Is it any clearer now?
If you mean that the last part of the sentence can also be validly translated into English as 'that today was his wife's birthday' then I completely agree with you. Although there are contexts in which 'is his wife's birthday' is correct, the past tense would be more generally used and is grammatically correct.
For the first part of the sentence I wrote 'He finally remembered ...' simply because that word order sounded natural to me in English, and was marked wrong. There's nothing wrong in English with the word order in the given answer ('Finally he remembered ...) but my answer should be accepted as well.