You will need to factor in getting off the plane, collecting your luggage, etc. After you are in the shuttle, it should take somewhere around 1 hour. Have you tried calling MidWilShuttle and asking them how long it usually takes?
I use Midwilshuttle. They pick up kerbside outside arrivals, which is so much easier.
Depending on which terminal you arrive at, it could take up to 30 minutes to leave LAX property, as its a one system. Terminal 1 is the fastest to get to but the longest to exit from.
Terminal 8 is the longest to get to but the fastest to exit from.
Once you exit LAX property it usually takes 40 to 60 minutes.
Landing at10.20am, you should be meeting your driver at about 11.15am, so I would estimate about 12.30pm you should arrive at Disneyland Hotel.