  We have performed the first numerical study of minimally doubled fermions of
the Karsten-Wilczek class in the quenched approximation. This requires fixing
the counterterms, which arise due to hypercubic symmetry breaking induced by
the Karsten-Wilczek term. Non-perturbative renormalisation criteria are
formulated after a detailed study of the parameter dependence of mesonic
observables. Minimisation of the mass anisotropy of the pseudoscalar ground
state fixes non-perturbative renormalisation conditions for the counterterm
coefficients. These anisotropies are mapped out by probing different euclidean
components of the transfer matrix through calculations of the pseudoscalar
ground state mass in different directions. The chiral behaviour of the
pseudoscalar ground state is studied with the tuned Karsten-Wilczek action for
multiple lattice spacings. Light pseudoscalar masses ($ M_{PS} \lesssim
250\,MeV $) were achieved in the quenched approximation without encountering
exceptional configurations. The presence of quenched chiral logarithms is
studied under the tentative assumption of Goldstone Boson-like behaviour.
