Bulldog's advice is right on. Three nights at Mammoth Lakes or at Rock Creek should be all the acclimatization you need.
A more primitive option would be to pack light -- backpacking gear should not be too difficult to transport on your trip -- and camp at least two nights at Horseshoe Meadows walk-in campground. There are no showers there, but Lone Pine is 30 minutes down the road, where the Whitney Portal Hostel has nice showers for $5. A shower and a restaurant meal, and you could head back up to spend another night at 10k.
Since you're coming from so far, I'd think that spending more time here would be your preferred choice, so taking day trips to White Mountain and the Bristlecone Pine forests, Yosemite's Tioga Pass and Tuolumne Meadows, and even the Bodie ghost town would give you a full plate of rewarding visits. Even Yosemite Valley is within reach.
Since you're looking to acclimate for Whitney, choose trips in order of altitude exposure -- lower elevations earlier, White Mountain would be the last (highest) place to visit.
You will definitely enjoy seeing those places!
Oh... and if you fly in and out of Las Vegas, you can drive across Death Valley and see yet another landscape jewel on your trip.
