Akichow- You're going to like the Miter Basin, and that's the perfect time to go. The route via Lower Soldier is good, and mostly on trails. To answer your question, yes, you can take Old Army Pass and easily connect with the New Army Pass trail on the west side. Lower Soldier Lake to Sky Blue Lake is a good one-day hike. From Lower Soldier climb the hill on the west-northwest side of the lake and from just over the top follow the contours (heading north), then dropping down gradually to meet-up with the use trail that heads up Rock Creek. The use-trail comes and goes in places, but the route is pretty obvious. At the head of the valley is a steeper part and a short ways up it, cross over to the west side of the creek to some ledges that will take you to Sky Blue Lake at its outlet. A good small campsite is located around to your left in the ledges by the lake, or further on the east shore. Good day trips from there are a Mount Mallory climb, Iridescent Lake, the lake up by Mount McAdie just south of Crabtree Pass, and Arc Pass.
Are you comfortable with off-trail route finding and good with map and compass? If so, a great alternative route is to go via Upper Soldier Lake: Camp the first night at Lake 4 and the next day go up Old Army Pass and head up the Crest toward Mount Langley. At around 12480' or so, peal off to the west and head down into the Soldier Lakes drainage. There is a use trail, but it's discontinuous and easy to loose, so take the time to navigate with the map. A really great camping spot is a big sandy area along the north side of Upper Soldier Lake. It's a good 1-day hike to Sky Blue from there. From Upper Soldier, follow the drainage down (you can pick-up switchbacks of an old trail - keep more to the right as you go down and you'll find them) to where a creek comes down from the right in a steep narrow gully and ascend that (the creek shows on the map on the southwest edge of the Major General; again, there is a use trail). You top-out at a small tarn and from there contour along the side of the ridge heading north and eventually drop down to the use-trail along Rock Creek.
For a paper map to carry along, the Tom Harrison Mt. Whitney High Country is good with up-to-date locations for the official maintained trails. But for off-trail route finding, it's always good to have the USGS 7.5 min sheets also. A couple of the ones in that area are metric so the contour lines don't exactly match-up with the ones that are in feet, but it's not really a problem.
You can exit from the Miter Basin the way you came in with the same campsite itinerary. (Or on the last day you can pass-by the camp in the Cottonwood Lakes and make for a long day of hiking out to the Cottonwood Lakes trailhead). I find that to summit Langley on the way out is more enjoyable because of being much better acclimatized and conditioned.
Steve - Lower Soldier Lake is just off the southwest ridge of the Major General. A short spur of trail leads to it from the Rock Creek Trail. Upper Soldier Lake is the upper one, directly east.
