Chances are the prednisone has a lot to do with it. It makes the dog thirsty so your dog drinks more and urinates more. It will improve when you taper off the prednisone. Right now, all your efforts at keeping her clean and dry are worthwhile.
I love the quilted pads, too, as a back up for diapering. She may still get her hip wet on a quilted pad, it is very good that you are cleaning and drying her. Urine burns on a down dog should not be underestimated, especially in a pressure area such as the hip. My dog was kept on plain disposable pads (like puppy pads) at the vet and they didn't keep his diaper on or catheterize him and he got a urine burn under his hip where he laid on a wet pad. I had to remove him from that vet. Those disposable pads are almost worthless, they keep the bedding dry but they will NOT keep the dog dry. When the urine hits the pad, it will gravitate to the lowest point of the bed, which is usually under the hip. Something like the fake sheepskin they put in whelping boxes would do a better job of keeping the dog up out of the urine, letting it trickle through.
Did they say anything about expressing her bladder? Here is a link describing it.http://www.handicappedpets.com/mediawik ... dog_or_cat
Is she urinating in big puddles, or is she kind of dribbling a lot all the time? Usually if a dog is really paralyzed, you need to express the bladder in order to empty it completely every 8 hours. Otherwise they get a urinary tract infection, which can be serious or even fatal if not treated. The bladder may also stretch from being overly full all the time, and then even if the dog recovers the nerve function necessary for bladder control, she will not be able to urinate normally because her bladder has stretched.
Sometimes people think that because they see the dog urinating, it means the dog is releasing her bladder voluntarily. However with a paralyzed dog, it may be that you are seeing overflow, where the bladder fills to capacity, then releases some of the urine, which looks like the dog is peeing. In that case, the bladder is not fully emptying and stale urine remains in it all the time creating an environment for germs to grow, and the dog gets a bladder infection. If your dog always urinates before you take her outdoors, you may want to try expressing her bladder while she is lying down. For that, you would put something absorbent under her bottom and press on her bladder to help empty it. There is a description of expressing a dog who is lying down in the link above.
What kind or size of a dog is she?