At the end of The Long Dark Tea-Time of the Soul, the following things resolve Odin's troubles:
Thor changes an eagle back into a fighter plane, which (unintentionally?) kills the Draycotts, whom Odin entered a contract with.
Odin re-entered the hospital he wanted to stay in, on more favorable terms.
A new Guilt God emerged, apparently from Dirk's old fridge, and dealt with Toe Rag and the green-eyed giant, as they were escaping from Valhalla.
Apart from getting rid of his old fridge, and possibly giving an old god (who was perhaps a guilt god, as Dirk felt a "nameless roaring guilt that came howling at him out of nowhere" at their encounter) a coin, which apparently helped precipitate the 3rd item above, I don't see that Dirk did anything for the resolution of this story. But as Toe Rag and the giant were escaping anyway, this doesn't seem crucial to the resolution.
Dirk made his way to Valhalla, and tried to help Odin against the Draycotts, but quickly gave up. Kate apparently is responsible for 2, and Thor for 1. As far as I can tell, it seems like he figured out, more or less, what was going on, and tried to help fix things, but without success in any major way. Did Dirk actually do anything useful for the end of this story? (Am I missing the significance of 3?)