+1 Most LAX-IAD nonstops land after 4 PM on the East Coast, giving OP under 4 hours, allowing for time to get from the mid-field gates to the main building and outside, and reversing + TSA

The whole family has been awake since 5 AM @ LAX, breakfast (maybe not for the still sleeping kids) & ride out to the airport, luggage & checking in, security & out to the gate for pre-boarding with the kids - munching on whatever in flight.

Getting around from IAD is either via expensive taxi as the train/bus/Metro transfer is just not practical for 3 or 4 together - unless OP has been to DC area before & comfortable with a car rental to drive around to sightsee (errr, the rush hour traffic on I-66 HOV toll road isn't exactly fun either)

Either take a late morning departure with enough slack to not misconnect at IAD, or, arrive early/mid-afternoon - get a day room near IAD with free shuttle services, check in and relax, take a shower, have a late lunch/early dinner or whatever, watch TV, swim if there's an indoor pool or take a nap and just relax, then - go back to IAD for the continuation segment around 9'ish.

Or, skip the hotel day room idea and head to 1 of the 2 nearby mall via taxi -

Reston Town Center - 50 stores, 30 restaurants & move theater, etc.
Distance from IAD: 6.0 mi http://www.restontowncenter.com/

Dulles Town Center - 185 stores, 5 department stores, 23 fast food and 10 fine restaurants, etc.
Distance from IAD: 7.5 mi http://www.shopdullestowncenter.com/