how can an adjective function as an adverb
Solution 1:
From a comment by BoldBen...
The word "happy" is not modifying "goes" it is modifying "she". In other words it is an adjective describing the patient's state of mind. There is, actually, a comma missing from the text; the whole passage from "because" to "later" is parenthetical so the main part of the sentence is "Yet she goes home happy." This makes it much clearer that the adjective refers to the woman.