The effects of the external intermittent behaviour on the Kolmogorov constants $C_{k1}$ and $C_2$ in spectral and the physical spaces are investigated using high-resolution direct numerical simulations of a turbulent plane jet. Well-defined $- 5/3$ energy spectrum and $2/3$ structure function can be found in the intermittent flows without large-scale vortex shedding. For different cross-wise positions, the profiles of conditional energy spectra and conditional structure functions exhibit self-similarity at small and intermediate scales when normalised by the conditional Kolmogorov scale of the turbulent region. The conditional Kolmogorov constants are close to those of the fully turbulent flow. The constants $C_{k1}$ and $C_2$ are found to have a power-law dependence on the intermittency factor $\gamma$, that is, $C_{k1}\sim \gamma ^{1/3}$ and $C_{2}\sim \gamma ^{1/3}$, except for the scaling of the structure function in the highly intermittent region with $\gamma =0.25$. In the highly intermittent region, e.g. $\gamma =0.25$, the scaling in the conditional structure function can be considerably influenced by the blocking/sheltering mechanisms of the turbulent/non-turbulent interface (TNTI), leading to slight deviations from self-similarity. We further confirm that the conditional structure function recovers self-similarity after excluding a turbulent region at an average distance of approximately $20$ Kolmogorov length scales from the outer edge of the TNTI, which is comparable to the mean thickness of the TNTI. These findings contribute to the modelling of the edge of a turbulent region.