There are several intriguing similarities between the Jonah account and what happened to Jesus, though it is understood that Jonah was initially disobedient to God, while Christ never was. However, consider these points:
In the tempest, the sailors fired one question after another at Jonah. "Who are you?" "What is your mission?" Death faced Jonah for his honest answers, and he knew it. Jesus was questioned, and he knew death faced him no matter how he answered, but he never lied. Jonah had lots cast over him. So did Christ. Jonah was given over to death, that the sailors might live. Christ was sacrificed that we sinners might live. The Lord's salvation was worked out through Jonah, and supremely through Christ.
Ah, but you note that Jonah never actually died while in the belly of the great fish, yet Jesus actually died. This is the point of your question, suggesting to you that Jesus never actually "completed" the sign of Jonah. The idea seems to be that Jesus should never actually have died if the parallel was to remain intact.
However, it is already obvious that the Jonah account was never meant to be an exact parallel for what would happen to Jesus. Jonah was trying to run away from God, to disobey God's will for him. That could never be said of Christ, who came to do God's will as written in "the book" about him. Jesus took Jonah to be a historic character with real events, but note how Jesus restricted himself in what parallels he drew out?
Jesus knew Jonah never died then, but Jesus knew that he would, therefore he never mentioned Jonah, or himself, being dead for [part of] three days. He stuck to being "in the heart of the earth". We all know Jonah was in the sea, not the earth. No attempt is being made by Jesus to exactly parallel events. The critically important point is that what befell Jonah was God's punishment on him for his sin, bringing him as close to death as possible without actual death. But with Christ, he would actually have to die as God's judgment and punishment on sin was poured out on him during the darkness of Golgotha.
Just as God prevented Isaac the son being sacrificed, providing another sacrifice, this pointed to Christ, his Son, as the ultimate sacrifice. Isaac was "a shadow" of Christ, teaching a massive lesson, yet Isaac only came as close to death as God would permit. Christ was the reality, and really had to die as a sacrifice, though he was without sin. So, with Jonah. He pointed symbolically to the coming Christ. Jonah only came as close to death as God would permit. He permitted the actual death of his Son, because those old testament events were 'types', illustrating some vital truths about God's plan of salvation. Not every detail had to be paralleled; indeed, they could not be. Let me conclude by quoting from this scholar's study-book on the Bible book of Jonah:
"The Lord wished Jonah's punishment to be as close as could be to that
of dying without actual death. Because of this Jesus was able to draw
out the parallel with his own death, involving the punishment for sin,
and also the fact that after the experience there was a resurrection
to come. This was to give his followers a frame of reference and hope
when they came to grapple with the reality of the cross...
...the definitive interpretive word of Jesus himself [bringing out]
various truths in the incident.. for instance, the heinousness of
rebellion against God, death as the appropriate penalty for this, the
fact that God is able to rescue from death, and that his grace and
salvation are capable of offsetting even the most atrocious of sins.
Indeed these were lessons that Jonah and his contemporaries were
expected to learn from the prophet's experience." Jonah, Micah,
Nahum, Habakkuk and Zephaniah, John L. Mackay, p.32, Christian Focus,
1998
Jesus surpasssed the sign of Jonah, for sinful Jonah was being punished for his own sin, while the Son of God was being punished for our sins.