。
She risked her own life to save the child's.
Literal
She [topic-は] her own life [object-を] stake [then/by-て] child's life [object-を] saved.
命をかける ('to stake / risk one's life') is a fixed collocation; the te-form 命をかけて links the two clauses with an implicit means-or-cause reading: she put her life on the line, and that's how the child was saved. The repeated 命を at parallel structural positions creates a rhetorical symmetry — her life balanced against the child's. The noun 命がけ ('at the risk of one's life') is derived from this same verbal collocation.