。
What is she so eager to know?
Literal
She [topic-は] what [object-を] so-much know-wants is [I-wonder-のか].
Two patterns combine. ~たがる ('want to V,' for third-person desires) is Japanese's standard way to attribute wishes to someone else without claiming direct access to their inner state. Adding ~ている gives '(currently) showing the desire.' そんなに ('so, that much') intensifies. The closing のか is the explanatory + question particle, here functioning as a softly-mused question or self-question: 'just what is it that she wants to know so much?' Without polite verb endings, the line reads as the speaker's own wondering rather than a question put to anyone.