A picture of your plant may have helped with advice, but -

Prune anything broken or jagged to a clean cut with sharp pruners. Then, I think I'd wait until Spring when next years growth buds are beginning to show to do further maintenance. Prune to the shape you desire by cutting back to a pair of healthy buds. For now, you might want to consider one of the repellent sprays like Liquid Fence or my fav, Bobbex, to deter deer.

Should I prune the broken branches at the base, or at point of break? I'd prefer to have the branch grow in length rather than split at the broken point (they are rather short now).

For now, I'd prune at the break. You can do further shaping in latest winter or early Spring when your plant begins to bud out and you can see what it's doing for growth before guiding it with your cuts.

Clean sharp cuts are always better than jagged or torn, and less chance of a secondary disease or insect issue appearing.