An apparatus for removing broken fasteners from an engine cylinder head having a plurality of internally-threaded bores formed therein. The apparatus may comprise a guide plate having a first surface and a second surface, with the second surface being for positioning toward the cylinder head. The guide plate may have a plurality of apertures formed therein to correspond to bores of cylinder heads, and the apertures may extend through the guide plate from the first to the second surface. The apparatus may include a plurality of securing elements removably insertable into one of the apertures for removably securing the guide plate to the cylinder head. The apparatus may include a plurality of drill positioning sleeves removably positionable in the apertures of the guide plate. The apparatus may also include sleeve securing structures associated with each of the apertures to removably secure one of the drill positioning sleeves in the aperture.