"She declares the house is hers."

Translation: Cô ấy tuyên bố ngôi nhà là của cô ấy.

[KyleMunz]

What is the difference between ngôi nhà and just nhà?

[TehVanarch]

Nhà just refers to houses in general but can also be used as a prefix for certain occupations like nhà báo (journalist), nhà chính trị (also known as chính trị gia, politician). When counting houses you can use ngôi or căn.

[XieC2]

Ngôi is one of the valid classifiers for nhà. Nhà without ngôi would be awkward in this sentence (in the absence of another modifier like này) because of the somewhat hard to explain conditional preference in Vietnamese for disyllabic constructions.