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$6.1inthisline-of-sight,consistentwiththeendofpercolationatthisredshift.ThenarrowrangeofreionisationhistoriesthatwepreviouslyfoundtobeconsistentwithawiderrangeofLy$\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