There are many research works done in the field of human falls recognition, but, to the best of my knowledge, none of them is available in OpenCV. You could Google it, to find some interesting papers.
A HOG detector on human won't work, because it has been trained to recognize human walking or standing still. What you are looking for is a kind of human action recognition. You could train a new HOG detector with people on the floor, but this isn't really a fall detector, more a "people on the ground" detector. This depends on your application/motivation.
As you are using a Kinect, I strongly suggest using an approach that takes advantage of the skeletons you could extract. A simple search on Google give me a lot of results. Have a look at the Microsoft Datasets page here. Look also at the papers that cite these datasets, they are more "actions recognition" oriented, but they could give you some clues to your problem. (if you are interested by a solution easy to implement, and dealing with skeletons without training a SVM, or anything else, look at this paper which is mine…;-) but many others interesting papers are available on Internet.)
Yes, you should train another detector that is able to detect those type of humans (falling ones). Or try to rotate the frame if it does not take too long