Your route is the longer of the two by a wide margin. The southern terminus of the Sasse Mountain Trail offers the quickest approach. You should add this info in to the page some place:
South Route (Sasse Mountain Trail)
From Cle Elum River Road take a right (east) on FR-4305 just south of Wish Poosh Campground. Follow the signs up to road's end at the trailhead (c. 3,600 ft). There are two turns getting there (both have signs pointing the way).
The trail is a motorcycle or bicycle route mostly but can be done on foot rather easily. Simply hike the winding track all the way to the summit. Distance is about 2.5 miles and gains (including ups and downs) about 1,500 ft. Allow 1 hour going up and 45 minutes going down (or faster if you jog it).