A few years ago, I tried getting to Sky Blue Lk via Mt Langley, but I didn't carry my pack high enough before dropping it to summit Langley. (You need to climb to 13,600 or higher on Langley before you could descend the west slope. I think Secor calls the west slope of Langley a Class 2 walk)
So I descended to upper Soldier Lk, planning to work my way north and northeast around the nose of Major General. I have been told that route works. ...but again, I didn't go high enough, and got into a bushwhack descent down to Lower Soldier. I never saw anything inviting to climb northward in the south-facing stream trace between upper and lower Soldier, and continued to the n-nw shore of lower soldier, where there is an easy ascent to the ridge nw of lower Soldier Lk.
I proceeded north cross-country, and eventually reached Rock Creek. I finally made to the north shore of Sky Blue, at dusk. I was so exhausted that I lost all my dinner.
I had camped the night before at the northern-most unnamed lake of Cottonwood lakes (gps 36.51367, -118.22725. elev 11,600 )
, and climbed the steep slope to the Langley crest just to the west.
So... good luck on your LeConte trip. If I were to try it again, I'd skip Langley, go to upper Soldier, and skirt the nose of Major General. If you're able to carry your overnight gear nearly to the Langley summit, you could try that route--I am/was too old for that route.