Dryfly,
Whenever I'm doing a day hike of Whitney, I'll hike to Lone Pine Lake (round trip 5 miles and 9,900 feet elevation). I'll stay there for 3 hours and fish, sunbathe, and enjoy the scenery. I've done this the past two times and have reached the summit both times.
If you don't feel like hiking that day you can drive up to or spend the night in the campgrounds at Onion Valley or Horshoe Meadows, both are almost at 10,000 feet elevation. But, staying at WP is fine too. Good luck. I'll be doing a day hike of Whitney on July 1. Acclimating my above schedule on June 29 and a rest day June 30, then up to bag two 14ers (Muir and Whitney). Best of luck to you. Hope this helps.
Greg Lukenbill a.k.a. GLuke