I searched for it with my life on the line and found it.
Literal
Life-stake [with-で] found-out.
命がけ (命 + 掛け, 'life-staking') is a set noun meaning 'risking one's life / with utmost effort.' Used adverbially with で. 探し出す is a compound verb (探す + 出す) meaning 'to seek out, find by searching.' A dramatic flourish — 'I hunted this thing down at any cost.'