You do understand, Dennis, that trekking to Everest Base Camp isn't exactly like traipsing off to ride Space Mountain at Disney World, right? One piece of good news for you is that there doesn't seem to be much evidence that kids are more prone to altitude sickness than adults. But let's take the best-case scenario—that you're using the name Dennis as an alias, and you're actually Ed Viesturs. Your ten-year-old daughter (Edwina?) is a chip off the old chiseled block, and—in fact—she came back just last night from bagging El Capitan. Solo. In a day. Using only her pinkies.
Regardless, you, as a responsible parent, are going to make sure that Edwina acclimatizes slowly and properly before reaching base camp at 17,575 feet. This means starting the expedition at 9,350 feet in the village of Lukla, and taking ten days or more to reach your ultimate destination. The trek's total duration would be close to three weeks. Yes, the views along the way are enough to take what's left of your oxygen-starved breath away. But the much-traveled route makes the Appalachian Trail here in the States seem like a quiet, lonely path. And how much will reaching Everest Base Camp mean to Edwina, or any ten-year-old, when there are other, potentially more spectacular (and maybe memorable) Nepal treks? These are questions only you can answer.
If I were trekking in Nepal with my six- and eight-year-old kids (which would first require surgery to remove the complaining lobes in their brains), I'd opt for Part B of your question and head onto some part the 155-mile Annapurna Circuit first. The range in elevation is about the same as the hike to Everest Base Camp—varying from 9,200 feet to 17,500 feet—but you can more easily custom-build your itinerary. And on this route Edwina will find the Himalaya views more varied, the cultural experience more hands-on, and the crowds on the trail—even with the new road construction that's going on—generally sparser.
For guide services, Mountain Madness leads Everest Base camp trips for $3,675 per person, and Kathmandu-based Crystal Mountain Treks, operates half-circuit treks of the Annapurna Circuit for $1,845. In both cases, you'll need to consult with the companies regarding the age and abilities of your daughter.