I just got back from Ellensburg; here are the directions to navigate through town to connect with the John Wayne Trail.
Coming from the West, the John Wayne Trail ends at Water St and 15th Ave. Turn right on Water St and then left on 14th. Fourteenth takes you through the University Campus. Turn right on Alder St and continue past University Way (traffic signal). The trailhead is about a block and a half past University Way on the left. Directly ahead are the Fairgrounds which have public restrooms that should be open during the summer.

Let me know when you're coming through and maybe we can meet at the South Cle Elum Depot for coffee or lunch.