17:17a Later, the son of the woman who owned the house became ill,

After some time had passed, the son of the woman whose house it was became ill.
-or-
But there came a day when the son of the widow/woman whose house Elijah was staying in got very sick.

17:17b and his sickness grew worse and worse, until no breath remained in him.

He became more and more sick/ill, to the point that eventually he stopped breathing.
-or-
His condition/sickness kept getting worse until he died.

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