We present a detailed study of a double-peaked Ly α emitter, named LAE-11, found in the proximity zone of quasar J0910-0414 at z  ∼ 6.6 at a proper distance of d _QSO  ∼ 0.3 pMpc from the quasar. We use a combination of deep photometric data from the Subaru Telescope, Hubble Space Telescope , and James Webb Space Telescope with spectroscopic data from Keck/DEIMOS, JWST /NIRCam WFSS, and JWST /NIRSpec MSA to characterise the ionising and general properties of the galaxy, as well as the quasar environment surrounding it. Apart from Ly α , we detect H β , [OIII] _{λλ 4960, 5008} doublet, and H α emission lines in the various spectral datasets. The presence of a double-peaked Ly α profile in the galaxy spectrum allows us to characterise the opening angle and lifetime of the central quasar as θ _Q  > 49.17° and t _Q  > 3.8 × 10 ⁵ years, and probe the effect of the quasar’s environment on the star formation of the galaxy. LAE-11 is a fairly bright (M _UV = −19.90 _−0.12 ^+0.13 ), blue galaxy with a UV slope of β = −2.61 _−0.08 ^+0.06 and a moderate ongoing star formation rate (SFR _UV  = 5.55 ± 0.65  M _⊙ yr ⁻¹ and SFR _{H α}  = 12.93 ± 1.20  M _⊙ yr ⁻¹ ). Since the galaxy is located in a quasar-ionised region, we have a unique opportunity to measure the escape fraction of Lyman continuum photons using the un-attenuated double-peaked Ly α emission profile and its equivalent width at such high redshift. Moreover, we employ diagnostics which do not rely on the detection of Ly α for comparison, and find that all tracers of ionising photon leakage agree within 1 σ uncertainty. We measure a moderate escape of Lyman continuum photons from LAE-11 of f _esc ^LyC  = (9 − 43)%. Detections of both H α and H β emission lines allow for separate measurements of the ionising photon production efficiency, resulting in log( ξ _ion /Hz erg ⁻¹ ) = 25.57 _−0.12 ^+0.16 and 25.63 _−0.11 ^+0.17 for H α and H β , respectively, when using the median f _esc ^LyC . The total ionising output of LAE-11, log( f _esc ^LyC ξ _{ion,H α} /Hz erg ⁻¹ ) = 24.25 ^+0.26 _−0.29 , is higher than the value of 24.3 − 24.8 that is traditionally assumed to be needed to drive reionisation forward.