Using binomial theorem

$\large (\frac{x^{18}}{k^{17}}+\frac{k^{16}}{x^{15}})^{19}$

General term: $_{19}C_{r} \times (\frac{x^{18}}{k^{17}})^{19-r} \times (\frac{k^{16}}{x^{15}})^{r}$

$=_{19}C_{r} \times (x^{342-18r})(k^{-323+17r}) \times (k^{16r})(x^{-15r})$

$=_{19}C_{r} \times (k^{-323+33r})(x^{342-33r})$

Put $r=10$ :

$_{19}C_{10} \times (k^{-323+33(10)})(x^{342-33(10)})$

$=92378(k^{7})(x^{12})$

then, $92378(k^{7})=92378$

$k=1$