。
She asked me whether I'd be free this Sunday.
Literal
She [topic-は] this-coming Sunday [topic-は] free [whether-or-not-かどうか] me [to-に] asked.
~かどうか packages an embedded yes/no question into a noun-like clause that can be plugged into a verb like 聞く, 知る, or here たずねる ('to inquire'). The structure preserves the embedded question's polarity: 暇かどうか asks 'free or not.' 今度の can mean 'this coming' or 'next' depending on context; with future-time 日曜日 it's the upcoming Sunday. The は after 日曜日 signals it as the topic of the embedded question — 'as for this Sunday, free or not.'