Kelly, it's too bad about your friends backing out. Unfortunately, that happens all too often.
And be nice to other hikers -- call the Wilderness Permit number (760-873-2483), and release those empty slots. Others will then be able to get them and plan their trips.
But you won't be alone on the trail. The daily quota is 100 day hikers and 60 backpackers. I'd suggest you just go by yourself, and hook up with the other hikers you meet on the trail.
...And I just saw your Shout Box question:
> How early should I get started if I plan on doing Whitney in one day? Relatively fast hiker...
Some people start around 3 AM (a few earlier!). If you're fast, plan on about 6 hours to summit.