Angle's measure

As angles of the same segment are equal $\angle HGK = \angle KIJ$ and $\angle GHK = \angle KJI$ we conclude that $\triangle HKG \ \ and \ \ \triangle JKI$ are similar. Therefore $JK = \frac{IJ}{GH} \cdot HK$ , i.e. JK = 6. With sides 6, 8 and 10 $\triangle JKI$ has $\angle JKI = 90^\circ$ .

Notice that triangle $HKG$ is a $3-4-5$ right triangle, therefore $\angle HKG=90^\circ$ . That means $\angle JKI=\boxed{90^\circ}$ .