‘And YHWH prepared a great fish to swallow up Jonah, and Jonah was in the belly (innards) of the fish three days and three nights.'
Meanwhile YHWH had not forgotten YHWH and as God of land and sea had already made provision for Jonah by arranging for a large fish to be in the area, so that as Jonah began to drown in the turbulent seas, the fish might swallow him. And when it did he was inside the fish for ‘three days and three nights', which in Israelite terminology indicated ‘a day or two'. (A ‘day and a night' could refer to part of a day, seeing it as part of the day and night cycle. Compare Esther 4:16 with Esther 5:1). He had not deserted His prophet, but had arranged for his rescue. The word for ‘belly' simply means the innards, and is not necessarily specifically referring to the whale's stomach. They did not know the physiology of whales.
Jonah, who had found himself drowning in the sea, and being dragged down into the depths, was, once he found himself alive and well and able to breathe in what appeared as some kind of chamber, grateful to God, and the psalm in chapter 2 expresses his gratitude. He probably did not quite know what had happened to him, or where he was (he would find that out later), but he knew that he was alive and was therefore confident that if he repented God intended to spare his life. Ironically he found himself in the same position as the Ninevites to whom he had refused to go, as one who was under sentence and deserving of death, but with an opportunity of repentance. The Psalm adequately expresses this position, and is a necessary part of the story. Without it there would be no indication of Jonah's repentance.