Its about a 45 min drive to Legoland from Anaheim, I would not suggest using a private driver! that would be expensive.
Since you want transport only, and not a Tour with a ticket, you would be able to get a Transport only option from Anaheim to Legoland with Graylines 1 ride Transportation Pass. Children however are not free with this transport.
You might compare that cost with something like taking the Train and then a Taxi from Anaheim to the Legoland Hotel to see what the costs may be, but it may work out to be about the same since I dont believe the Train stops that near to Legoland.
You may also want to look at the discounts for Legoland shown on the Mousesavers website which is a good resource for you as the list the various current ones ( they dont sell tickets) and there are usually quite good ones for Legoland.