if i may a few fixes:
at the end change the line
http.ListenAndServe(":3000", router)
with
http.ListenAndServe(":3000", handler)
and the api as by convention it should be more like:
router.HandleFunc("/healthz", Healthz).Methods("GET")
router.HandleFunc("/todos/completed", GetCompletedItems).Methods("GET")
router.HandleFunc("/todos/incomplete", GetIncompleteItems).Methods("GET")
router.HandleFunc("/todos", CreateItem).Methods("POST")
router.HandleFunc("/todos/{id}", UpdateItem).Methods("PUT")
router.HandleFunc("/todos/{id}", DeleteItem).Methods("DELETE")
this is a very good article on how to REST API should be written