Context. Observationally constraining the atmospheric temperature-pressure (TP) profile of exoplanets is an important step forward for improving planetary atmosphere models, further enabling one to place the detection of spectral features and the measurement of atomic and molecular abundances through transmission and emission spectroscopy on solid ground. Aims. The aim is to constrain the TP profile of the ultra-hot Jupiter KELT-9b by fitting synthetic spectra to the observed H$alpha$ and H$beta$ lines and identify why self-consistent planetary TP models are unable to fit the observations. Methods. We construct 126 one-dimensional TP profiles varying the lower and upper atmospheric temperatures, as well as the location and gradient of the temperature rise. For each TP profile, we compute transmission spectra of the H$alpha$ and H$beta$ lines employing the Cloudy radiative transfer code, which self-consistently accounts for non-local thermodynamic equilibrium (NLTE) effects. Results. The TP profiles leading to best fit the observations are characterised by an upper atmospheric temperature of 10000-11000 K and by an inverted temperature profile at pressures higher than 10$^{-4}$ bar. We find that the assumption of local thermodynamic equilibrium (LTE) leads to overestimate the level population of excited hydrogen by several orders of magnitude, and hence to significantly overestimate the strength of the Balmer lines. The chemical composition of the best fitting models indicate that the high upper atmospheric temperature is most likely driven by metal photoionisation and that FeII and FeIII have comparable abundances at pressures lower than 10$^{-6}$ bar, possibly making the latter detectable. Conclusions. Modelling the atmospheres of ultra-hot Jupiters requires one to account for metal photoionisation. [abridged]
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