Sounds like my cat. Both of my cats had bladder infections a few years ago and were peeing everywhere. After the bladder infections were cleared up one cat was back to just the box. My other cat, however, continued to pee on beds, bath mats, carpeting and still use the litter box, also.
The first thing is you have got to do is get ALL of the smell gone. That may mean throwing out the kitty bed, unless it is completely washable. Buy one of the odor neutralizers just for cats, because cat urine is far more pungent than dogs, as you know! If you're very lucky and can completely rid the area of the urine smell the cat might only use where she smells urine, i.e., the litter box.
If you're unlucky, like me, it becomes habit. I only have carpet in my closets, so I keep all closets closed and bedroom doors stay closed unless I'm in the room to keep an eye on things.
My cat has gotten better, but her motto is "I'll be good -- until I forget!"
L. F., "mom" to two cats and a VERY understanding husband!