Polynomial time algorithm to find an induced cycle in non-chordal graph
Since I found a solution myself, I'm gonna post the idea. It suffices to find chordless paths. For a vertex $v$ and $u$ its neighbour, we can delete the edge $\{u,v\}$ from $G$ and find a chordless $u$-$v$ paths. Connecting these will result in a chordless cycle.
A very simple algorithm finding such paths is described here. Naive algorithm runs in time $O(n(n+m))$ per $u$-$v$ path, they further give more efficient algorithm with running time $O(n+m)$ per $u$-$v$ path.