Drive from LA to the Grand Canyon, stay overnight at Grand Canyon for 1-2 nights and then drive to Las Vegas. You will be able to visit Hoover Dam on the drive from Grand Canyon to Las Vegas.

LA to the Grand Canyon is 7-8 hours, just drive it in one day. Days are getting fairly short at that time of year, about 10 hours of daylight which is why I would recommend you spend 2 nights there as you won't have time to see much the first day you arrive. The link below is a calendar of sunrise/sunset times at the Grand Canyon (also moonrise/moonset which may come in handy because there's a full moon on December 6th and seeing the canyon in the moonlight could be awesome).

http://tinyurl.com/kjv87en

It will be cold at the Grand Canyon in December, with possibly some snow on the ground. There is a fairly level trail along the rim which should be easy enough for you. They also have a shuttle bus route that would help if you aren't up to walking much.

If you want to visit more parks, Death Valley is a 2-3 hours drive from Las Vegas. Zion is about 3 hours away. Valley of Fire State Park is less than an hour from Vegas.

Oh - the overnight drive is one of tom's better suggestions . . . He seems to have a deathly fear of car rental drop off fees and gas prices, so he often advises multiple trains, coaches and transfers to get from a simple point A to point B. so driving throughout the night - he took pity on you

I would also do LA > Grand Canyon (you'd definitely need 2 nights) and on to Las Vegas. From LV you can do various day or overnight trips to National parks to the east and west.