How do I use 着呢 correctly?
The book is correct and your teacher is correct with their other suggested meaning.
KEY defines 着呢 as:

1 {colloquial} quite (intensifying particle following an adjective, e.g., yuǎnzhene 遠著呢/远着呢)
2 (following an action verb, stresses the action is ongoing [and unfinished]) ‖ (pronounced &quot;zhine&quot; in Beijing colloquial)

So if you see 着呢 after an adjective it probably means “very,” or “quite.” But, if you see it after a verb it’s most likely telling you that something is still ongoing.