Have you considered Amtrak via Sleeper Car service? You would take the Lake Shore Limited to Chicago, then you could take either the Southwest Chief to LA, or the California Zephyr to San Francisco and the Coast Starlight from San Francisco to LA.
There could be a day or two of sight-seeing in Chicago or San Francisco on either route, to break up the 3 or 4 days of train travel. The Walt Disney Family Museum in San Francisco is
NOT to be missed for any Disney fan. And the Coast Starlight route along the ocean from LA to San Francisco is one of the most beautiful train routes in North America.
Or, from Chicago you could take the Empire Builder to Seattle or Portland, then the Coast Starlight down from the Great Pacific Northwest to LA. You'd transfer trains in LA to catch an hourly Surfliner train to the Anaheim station, about a five minute cab ride from Disneyland and its hotel district.