Multidrug-resistant Klebsiella pneumoniae is an increasing cause of infant mortality in developing countries. We aimed to develop a quantitative understanding of the drivers of this epidemic by estimating the effects of antibiotics on nosocomial transmission risk, comparing competing hypotheses about mechanisms of spread, and quantifying the impact of potential interventions. Using a sequence of dynamic models, we analysed data from a one-year prospective carriage study in a Cambodian neonatal intensive care unit with hyperendemic third-generation cephalosporin-resistant K. pneumoniae. All widely-used antibiotics except imipenem were associated with an increased daily acquisition risk, with an odds ratio for the most common combination (ampicillin + gentamicin) of 1.96 (95% CrI 1.18, 3.36). Models incorporating genomic data found that colonisation pressure was associated with a higher transmission risk, indicated sequence type heterogeneity in transmissibility, and showed that within-ward transmission was insufficient to maintain endemicity. Simulations indicated that increasing the nurse-patient ratio could be an effective intervention.