やり遂げた is the past tense of やり遂げる. It is a compound verb which has two verbs in one word. The first verb is やり which is the stem of the masu-form やります meaning “to do” and the second one is 遂げる which means “to achieve” or “accomplish” so やり遂げる means “to see something through“ or “to bring something to completion”. It is usually used for something that is hard to accomplish.
よ is a sentence ending particle that is used when the speaker is telling something that the listener doesn’t know. So in this scene, Honoka is telling that they did it to her past self who said “let’s see this through” in her flashback.
最後 means “last” or “end”.
まで means “until”.
So 最後まで means “until the end”.
As you know, in Japanese, a verb usually comes at the end of a sentence so it’s supposed to be 最後までやり遂げたよ！ But you can change the word order when you want to emphasise, make the sentence catchy or make it sound more emotional.