  We present a study of the $S=-1$ meson-baryon interaction, employing a chiral
SU(3) Lagrangian up to next-to-leading order (NLO) and implementing
unitarization in coupled channels. The parameters of the model have been fitted
to a large set of experimental scattering data in different two-body channels,
to threshold branching ratios, and to the precise SIDDHARTA value of the energy
shift and width of kaonic hidrogen. In contrast to other groups, we have taken
into consideration the $K^- p\to K^+\Xi^-, K^0\Xi^0$ reaction data, since we
found in a previous work to be especially sensitive to the NLO parameters of
the chiral Lagrangian. In the present work we also include the Born terms,
which usually have very little effect, and find them to be non-negligible in
the $K^- p\to K\Xi$ channels, correspondingly causing significant modifications
to the NLO parameters. We furthermore show that the importance of the Born
terms becomes more visible in the isospin projected amplitudes of the $K^-p \to
K\Xi$ reactions. The measurement of processes that filter single isospin
components, like the $K^0_L p \to K^+ \Xi^0$ reaction that could be measured at
the proposed secondary $K^0_L$ beam at Jlab, would put valuable constraints on
the chiral models describing the meson-baryon interaction in the $S=-1$ sector.
