。
She has a great interest in housework.
Literal
She [topic-は] housework [in-に] greatly interest [object-を] is-holding.
興味を持つ ('have an interest, take an interest') is the standard expression for being interested in something, and it takes the object of interest with に: ~に興味を持つ. Add ~ている for the resulting state of having that interest, and you get 興味を持っている ('is interested in'). 大変 functions adverbially here ('greatly, very much'), modifying 興味. The whole sentence reads as a slightly formal/written description of someone's enthusiasm.