  High-signal to noise observations of the Ly$\alpha$ forest transmissivity in
the z = 7.085 QSO ULAS J1120+0641 show seven narrow transmission spikes
followed by a long 240 cMpc/h trough. Here we use radiative transfer
simulations of cosmic reionization previously calibrated to match a wider range
of Ly$\alpha$ forest data to show that the occurrence of seven transmission
spikes in the narrow redshift range z = 5.85 - 6.1 is very sensitive to the
exact timing of reionization. Occurrence of the spikes requires the most under
dense regions of the IGM to be already fully ionised. The rapid onset of a long
trough at z = 6.12 requires a strong decrease of the photo-ionisation rate at
z$\sim$6.1 in this line-of-sight, consistent with the end of percolation at
this redshift. The narrow range of reionisation histories that we previously
found to be consistent with a wider range of Ly$\alpha$ forest data have a
reasonable probability of showing seven spikes and the mock absorption spectra
provide an excellent match to the spikes and the trough in the observed
spectrum of ULAS J1120+0641. Despite the large overall opacity of Ly$\alpha$ at
z > 5.8, larger samples of high signal-to-noise observations of rare
transmission spikes should therefore provide important further insights into
the exact timing of the percolation of HII bubbles at the tail-end of
reionization
