We did a Monument to Anaheim road trip a few years ago (about 17 hr total drive time each way).

Hello, neighbor - we are also in Monument
OP - We've done the drive from Monument. We go straight through in one day. Once you add the cost of hotels, it starts to negate the savings on the airfare. We usually save upwards of $700 by driving and we like having our car there. The first time, we found airfare for $160 ea and my youngest was still a lap child - THAT was worth flying. I've never done the southern route, but the northern (out I 70) was not a bad drive at all IMHO. We do have family in CA, so we stay in Hesperia, which knocks a couple hours off the drive. We don't go directly to DLR, we're usually with family a couple days first, so I can't speak to if everyone would be ready to hit the park the next day.
Here are things that work for us on long drives with little ones. Maybe some of them will work for you.
- Leave early. Before breakfast. We drove three hours with them in their pjs and made the first stop their morning routine with a twist. Get dressed, have breakfast at a café... By telling them we were driving to <wherever>, we could hold off on getting into some of the time killers until we'd already put a few hours behind us.
- DVDs. We are not generally a DVD-in-the-car family. We only break it out for occasions like this. We have one that straps on to the headrest. I prefer it over tablets and the like because my oldest gets carsick if he's looking down (this was discovered on our first long drive, so you may want to practice run some of the things you're bringing so you can change plans ahead of time if you have to). We don't start the first DVD until we're at least an hour into the drive. The first hour is the easiest, especially if it's still dark. They think it's exciting and we KNOW they can make it to Denver with no issues.
- Change it up. We do not do back to back DVDs. We do two hours of screen time and a half hour to an hour of drawing, talking, singing, looking out the windows, whatever, between stops. They know that a movie and some time of something else means we're almost to the next stop. We also want them to actually *see* some of the country that we're driving through.
- Bring presents. Small ones. Just something that's new to them. Sometimes it's a small toy, sometimes a new DVD or book. Don't underestimate the value of novelty! We use them as rewards for good behavior. Bonus - we get two more hours of quiet while they "discover" the new toy!
- Bring snacks. Our kids don't get "junk" snacks very often. Packing up the car with things like chips and beef jerky and cookies makes it special and exciting for them. They usually get one or two snacks on each "leg" in between real meals.
- Plan a longer stop somewhere in the middle - for us it is a park in St. George,
UT. An hour or two to run around and play outside after lunch set them up for naps once we got back in the car.
- Do your best to keep them on schedule. We planned our timing around their routines as best as we could. If you know your child is normally super active around mid-morning, for example, plan to stop and let them run around at that time instead of trying to force them to sit and watch a movie. If they normally fall asleep after lunch, plan to stop for lunch right before the normal nap time, then they'll be a lot more likely to conk out in the car - and you don't have to wake them up to eat.
- Spending money as incentive. I think I read about this one on the DIS a couple years ago. If they are old enough to understand the value of money, it works pretty well. We gave them each a roll of quarters and every time they got out of control or too loud or fought, they had to pay us for listening to it. Whatever was left when we got there, they had to spend however they wanted. I think they lost about 2 quarters before all it took was me saying to my DH "hmm. do you think we need to be paid for this one?" and they'd be quiet
Hope you find at least one of these helpful and applicable to your own kids. Have a great trip - however you get there!