。
She finds inexhaustible enjoyment in music.
Literal
She [topic-は] music [in-に] inexhaustible enjoyment [object-を] is-finding.
尽きない ('inexhaustible, never running out') is the negative form of 尽きる ('to run out, be exhausted'); used attributively as here, it modifies a noun with a sense of bottomlessness. 見出す ('to find, to discover'), built from 見 + 出す, is more literary than the everyday 見つける — it suggests perceiving something hidden or latent. The whole sentence reads as elegant, written-flavored prose, the kind of sentence you'd find in a music critic's column.