All two-bridge knots are hyperbolic except for the $(2, k)$ torus links. For an “as simple as possible” (but still fairly difficult) proof, see Theorem 10.17 of Purcell’s book [Hyperbolic Knot Theory][1].


  [1]: https://www.google.co.uk/books/edition/Hyperbolic_Knot_Theory/iioDEAAAQBAJ?hl=en&gbpv=0