From the parking for the Longs Peak trailhead, begin hiking on the famous, well-worn and used Longs Peak Trail. The last time we were on this trail was 2007. At that time, there had been some major trail work on the first mile and a half of the trail with gravel having been brought in to raise the bed of the trail. Farther on however, the trail was still in bad need of repairs, especially as you get above timberline and the trail becomes entrenched in places. The trail initially heads WNW and there's a junction about a half mile up where you should stay left. In another mile, the trail comes to a series of a half dozen switchbacks, then crosses "Alpine Brook" which originates in the Jim's Grove area. Avoid taking the alternate trail to Jims' Grove (if it's still an open trail) and continue to tree line and the "Mills Moraine." The trail will come to an intersection. The right hand fork continues on to the "boulder field" and the Long's Peak route. The left fork takes hikers to Chasm Lake, at the foot of the east face "diamond" of Long's Peak. In 2007, there was a solar toilet nearby.
From this intersection you can head directly up the east ridge of Mt. Lady Washington. It's also possible to hike WNW along the Longs Peak trail a short distance and then turn west as well. Just choose whichever path looks most inviting to you. The first 200 feet of gain will be mostly on a tundra/embedded rock mix. Easy enough. As you continue hiking up, the angle of the slope steepens and the tundra gives way to more rock. The last 200 - 300 feet to the summit, the rocks gain boulder size and there's even stacked boulders that require scrambling around, through, over and under. This last part becomes a little slow and tedious.
Once on the summit, as mentioned before, the best feature of this hike is the close-up view of the Longs Peak east face. You can obtain an even better view by heading on over to the lower, west summit of Lady Washington, no more than 10 minutes away. Not only do you get to see the east face, but also Mt. Meeker and the "Ship's Prow" form a spectacular and imposing rock face to the south. Take your time if weather allows. Enjoy the view, eat some lunch, then return by the same route.
Mt. Lady Washington can be easily combined with Mt. Meeker but the addition in time to that route will be at least two hours, probably closer to 2 and a half or three because of the additional 1,600 feet in gain. Done alone, the Mt. Lady Washington hike can be done in a comfortable 8 hours round trip. Stronger climbers should be able to complete in much less time.
Links to other information, routes & trip reports for this peak that may be helpful.