  We show that if a hyperbolic knot manifold $M$ contains an essential
twice-punctured torus $F$ with boundary slope $\beta$ and admits a filling with
slope $\alpha$ producing a Seifert fibred space, then the distance between the
slopes $\alpha$ and $\beta$ is less than or equal to $5$ unless $M$ is the
exterior of the figure eight knot. The result is sharp; the bound of $5$ can be
realized on infinitely many hyperbolic knot manifolds. We also determine
distance bounds in the case that the fundamental group of the $\alpha$-filling
contains no non-abelian free group. The proofs are divided into the four cases
$F$ is a semi-fibre, $F$ is a fibre, $F$ is non-separating but not a fibre, and
$F$ is separating but not a semi-fibre, and we obtain refined bounds in each
case.
