She was looking for a room with a bath and found one nearby.

Literal

She [topic-は] bath-attached room [object-を] was-looking-for [explanatory-のだ] [but-が], this-nearby [at-で] found.

風呂付きの uses the suffix 付き ('attached/included') — a common way to describe amenities. のだが sets up background context before the main point.