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In this paper, we go further and propose a cosmological model-independent approach to simultaneously determine the Hubble constant and cosmic curvature with strong lensing time-delay measurements, without any prior assumptions regarding the content of the Universe. The data we use comprises the recent compilation of six well studied strongly lensed quasars, while the cosmic chronometer data are utilized to reconstruct distances via cosmographic parameters. In the framework of third-order Taylor expansion and (2, 1) order Padé approximation for for cosmographic analysis, our results provides model-independent estimation of the Hubble constant $H_0 = 72.24^{+2.73}_{-2.52} ~km~s^{-1}~Mpc^{-1}$ and $H_0 = 72.45^{+1.95}_{-2.02} ~km~s^{-1}~Mpc^{-1}$, which is well consistent with that derived from the local distance ladder by SH0ES collaboration. The measured cosmic curvature $Ω_k=0.062^{+0.117}_{-0.078}$ and $Ω_k=0.069^{+0.116}_{-0.103}$ shows that zero spatial curvature is supported by the current observations of strong lensing time delays and cosmic chronometers. Imposing the prior of spatial flatness leads to more precise (at 1.6$\%$ level) determination of the Hubble constant $H_0=70.47^{+1.14}_{-1.15} ~km~s^{-1}~Mpc^{-1}$ and $H_0=71.66^{+1.15}_{-1.57} ~km~s^{-1}~Mpc^{-1}$, a value located between the results from \textit{Planck} and SH0ES collaboration. If a prior of local (SH0ES) $H_0$ measurement is adopted, the curvature parameter constraint can be further improved to $Ω_k=0.123^{+0.060}_{-0.046}$ and $Ω_k=0.101^{+0.090}_{-0.072}$, supporting no significant deviation from a flat universe. Finally, we also discuss the effectiveness of Padé approximation in reconstructing the cosmic expansion history within the redshift range of $z\sim2.3$, considering its better performance in the Bayes Information Criterion (BIC).