This route description is based on the assumption that you do not have an adequate 4WD to make the drive into Governor Basin. Therefore, the mileage and elevation gain reflects a start from the beginning of the Governor Basin Road. If you are able to drive up to close to the Virginius Mine, the mileage will be reduced from 3.5 to about 1.5 and the elevation gain from 2650 to 1350.
Walk or drive up the Governor Basin Road (FR853.1C) to about a quarter mile shy of the Virginius Mine, where there's a nice flat area to park with nearby flower-strewn meadows. If you drove, you may want to consider leaving a few snacks around, away from your vehicle to distract the marmots from snacking on your vehicle while you're gone.
Follow the road on up to the mine and then hike south over steep rock to a bench at 12,800 ft. above a second mine. Continue across more rock, (or if you're early enough in the season and lucky, snow), and aim for the saddle east of the Mendota summit. To get to the saddle, you'll be hiking over a jumble of large, loose rocks. From this approach, the ridge that connects Mendota to T.5 looks intimidating with rocky spires, gendarmes and cliffs that seem to guard the way to the summit of T.5. Don't despair. Once at the saddle, an easy route unfolds through the obstacles as you turn left (east) and walk to the summit. Past the obstacles and Pt.13,337, the ridge widens out and becomes covered in small scree. Stroll on to the anticlimactic summit. Enjoy views down into Silver and Sydney Basins to the north & east and Marshall Basin to the south.
On the return hike you may want to stroll out along the narrow but easy ridge to the summit of Mendota Peak, even though it is unranked. The view is impressive with the huge vertical drop down to Telluride far below. As you return, there is an old trail that drops from the saddle north and takes a slightly more circuitous route back down to the main road. Snows in the earlier season may obscure much of this trail. Once back down into Governor Basin, take some time to enjoy the wildflower display. They are almost as abundant as Yankee Boy Basin and photographic opportunities abound with the spectacular St. Sophia Ridge as a rugged backdrop.