Basically, don't bother. Seriously, hear me out.
Route 66 doesn't exist and hasn't existed for such a long time now that its almost an irrelevence yet still us Brits are seemingly obsessed wtih driving it. There are bits left, notably in places like Arizona and California but for the most part all you will be doing is roughly tracing the route.
By driving from Chicago to LA in 14-18 days you are basically going to spend most of your time boring yourself to death driving through the featureless centre part of the US on interstates and highways thinking 'Route 66 used to sort of go near here'. This is rubbish.
Also, you are going to incur quite massive additional expense which you could otherwise have spent on cool stuff. The one-way car hire cost from Chicago to LA is going to be enormous, rental firms dont want cars to travel that far from the home depot if they can help it, so they penalise you with a huge drop fee and also by palming you off with the cars they want rid of when you come to collect. You are also going to find it less easy to get a decent flight deal with an open jaw return into Chicago and out of LA.
Instead, do a loop.
Fly into and out of LA. Rent something nice like a Mustang Convertible. Trace the route of the old 66 from LA towards Flagstaff, Arizona. This will get you almost all of the route 66 stuff you wanted anyway. Then you can hit the Grand Canyon, come back through Las Vegas, check out Death Valley, head up to San Francisco and then cruise back down the epic Pacific Coast Highway with the roof down in your Mustang convertible.
This trip will be 10x better and also cheaper. And you'll probably see just as much Route 66 stuff, but you'll also loads and loads of other great stuff, too.