Actually, there should be quotation marks (「奇跡は一度」) as he quoted what Byakuya said before when Ichigo took the lead in the fight the first time. So let’s look at that first.
kiseki wa ichido da. nido wa nai zo, kozō!
奇跡は一度だ: This is the basic sentence structure “A は B だ” meaning “A is B” so it means “a miracle is once”. So what he is saying is “miracles only happen once”.
二度はないぞ: 二度 means “twice”, は marks the topic, ない means “doesn’t exist” and ぞ is used to emphasise the statement in male speech. So he is saying “it never happens twice” or “there is no second time”.
So Ichigo quoted Byakuya’s line “奇跡は一度” meaning “miracles only happen once” and added だったよな.
だった is the past tense of だ so it means “was”.
よな is the masculine version of よね meaning “right?” which is used when you want confirmation from the listener.
So 奇跡は一度だったよな means “miracles only happen once, right?” or “miracles only happen once, was it?”.
じゃあ means “then”.
二度目はなんだ is again the “A は B だ” sentence but this time the B is なん meaning “what” and the A is 二度目 meaning “the second time” so it means “what is the second time?” so he’s saying “what do you call it the second time?”.
As you can see, they are only using simple sentence structures. Please note that だ is only attached to a noun or na-adjective, and if it’s an i-adjective or verb you just don’t need だ. Here are some examples. 明日は雨だ。meaning “It will rain tomorrow“. 図書館は静かだ。meaning “it’s quiet in the library“. 日本の冬は寒い。meaning “It’s cold in winter in Japan“.