I would just rent a car for the entire period.
I would bring my own GPS device, rent one from the company or use a app that provides live Traffic updates (I use a TomTom.)
As for getting from LAX to the Disneyland Resort, the most common route, and is the one used by professional transportation services is 105 east to the 605 south to the 91 east to the 5 south. The majority of the trip has carpool lanes (just 2 or more people in the vehicle are needed to qualify). To get to the GCH, use the special Disneyland Drive carpool exit, and make sure you do not use the flyover that goes straight to the Theme Park parking structure, stay to the right after getting off the ramp onto Disneyland Drive. Take Disneyland Drive down, and the GCH will be on the left, and has a traffic light to allow the turn. The average drive time is about 1 hour, maybe a bit less.
As for the return trip, lets work backwards, with a 1 PM flight, and I will presume domestic, the big issue is LAX and its usually long delays for the TSA security check, so we are looking at around 11 Am to be at the airport.
All rental car companies at LAX are off-site, so I would first drop off the family and luggage at the LAX terminal you are using, and then return the car solo, and take the shuttle back (Ti also works in reverse when getting the car). So add 30 minutes for the rental car return, and we are at 10:30 AM.
As mentioned, the average drive is about 1 hour, so I would add some time due to possible traffic delays, so about a 9 AM start sounds good.
Have a great trip.
The one issue would be Saturday January 13th, as that is a Marathon weekend, with a 10K run on Saturday. This will cause closures on Disneyland Drive due to the race, and that is the morning hours, so the return to LAX would be impacted, and I would add some extra time just to get out of the Hotel Parking Lot to the 5 freeway. I would also look at going south on Disneyland Drive to Katella, and make a left (east) to Haster, where you make a right, and then a quick left onto Gene Autry Way. This will place you to a special carpool entrance for northbound I-5.
Hope that helps.
EDITED to add- As for the Carpool lanes and the trip to/from LAX. the junction of the 91 and 5 freeway has a special carpool exit/connector that takes you directly to the left side of the freeway where they are located. The 91/605 junction,you need to get out of the Carpool lane and use the main ramps. The 605 doesn't have a carpool lane, but it is only a few miles of the trip (short segment). And the 105 has carpool lanes, but you need to get on the main lanes to enter/exit them. But they are a great help and recommend using them as much as possible. Also, the law is a bit weird, in Orange County, you have mainly open access carpool lanes (known as Diamond lanes), where you can enter/exit them most of the time, the lane is marked with "Elephant Tracks" which are wide white dashes (Aka fatter/larger than normal) . In Los Angeles County, there are only limited areas that you can enter/exit the carpool lanes. In both counties, you cannot cross the solid double white lines (Some are still yellow, but Federal law is making the switch to white. There will be signage overhead and attached to the K rail (median) that will mention upcoming exits just before the break in the solid lines And the break uses the Elephant Tracks to let you know it is OK to enter/exit.