White Mountain has a lot of snow and the road to the traditional start is closed. I suppose you could try the 10,000' vert from Bishop, but you could end up climbing 8,000' only to face post hole type snow.
Mt. Langley is also going to be a problem. The road, Horseshoe Meadows, is going to be closed for a while. The only real option would be from Tuttle Creek, which would involve 7,500' of gain and the NE Couloir which is full of 35 degree snow.
If you want a big day, I'd recommend hiking to Keynot or Inyo Peak. These are 11k+ peaks accessible from Lone Pine and are on the dry eastern side of the valley, i.e., the Inyo Mountains. I was up there a few weeks ago and I think you'll be good.
If you are deadset on the Whitneyzone, check out Thor Peak. It gets you a great view of Whitney, and it's only a 12er. If you take the southern approach (from Mirror Lake) you'll encounter some snow, but it may be OK. | 
