We compare a sample of five high-resolution, high S/N Ly$\alpha$ forest spectra of bright QSOs aimed at spectrally resolving the last remaining transmission spikes at $z>5$ with those obtained from mock absorption spectra from the Sherwood and Sherwood-Relics suites of hydrodynamical simulations of the intergalactic medium (IGM). We use a profile fitting procedure for the inverted transmitted flux, $1-F$, similar to the widely used Voigt profile fitting of the transmitted flux $F$ at lower redshifts, to characterise the transmission spikes that probe predominately underdense regions of the IGM. We are able to reproduce the width and height distributions of the transmission spikes, both with optically thin simulations of the post-reionization Universe using a homogeneous UV background and full radiative transfer simulations of a late reionization model. We find that the width of the fitted components of the simulated transmission spikes is very sensitive to the instantaneous temperature of the reionized IGM. The internal structures of the spikes are more prominant in low temeperature models of the IGM. The width distribution of the observed transmission spikes, which require high spectral resolution ($\leq $ 8 km/s) to be resolved, is reproduced for optically thin simulations with a temperature at mean density of $T0=(11000\pm 1600,10500\pm 2100,12000\pm 2200)$ K at $z=(5.4,5.6,5.8)$. This is weakly dependent on the slope of the temperature-density relation, which is favoured to be moderately steeper than isothermal. In the inhomogeneous, late reionization, full radiative transfer simulations where islands of neutral hydrogen persist to $z\sim 5.3$, the width distribution of the observed transmission spikes is consistent with the range of $T0$ caused by spatial fluctuations in the temperature-density relation.