I took my then 14yo son on a 2 night, 3 day trip from Cottonwood to Whitney portal via Mt Whitney a couple of years ago. The actual mileage is slightly under 39 miles, which for a 5 day trip would only be 8 miles/day. In fact, I think even your youngest might go crazy over 5 days. This is what it would probably look like:
N1 - Chickenspring
N2 - Rock Creek
N3 - Crabtree
N4 - Outpost
Cottonwood has become more popular as a point of entry for the Muir trail, so the 24 daily reservation slots are now gone for next week. However, CWP still has 16 spots available
each day as walk-in permits. To complement that 60/40 non-reserveable quota portion, there are 10 Whitney exit permits available
each day on a walk-in basis as well:
http://www.fs.usda.gov/Internet/FSE_DOC ... 143453.pdf
One last thing: the discussion above hasn't even addressed cancellations. There are always cancellations; SteveC over at the WhitneyZone compiles an annual list. I don't think there's more than 2-3 dates all summer long over the past few years that didn't have a cancellation.
My advice - for what's it's worth - is to go for it if your schedule, the weather and the family's motivation/health all line-up. The IAVC opens at 8am - just get there by 7:30am and take your spot before the door. There may be a few people in front, but the main herd shows after the doors have opened. The ranger may do a mini-lottery for the first 5-10 people in line to determine order of service for those without permits.
It would really be against the odds to (a) draw a bad ticket; and then (b) have everyone in front of you grab the remaining available walk-in permits for your exact dates & itinerary. If it did somehow happen, there are always alternatives. The USFS staff aren't there to bum your scene; they'll work to get you on the trail.