She is obviously aware that it is a bad thing. She typically hides under the couch until the next day, and she'll actively avoid us if she tries to go out for food and water. The first few times she did it, I brought her over to show her that pooping on the bed was wrong, but I don't know if it left the right impression.
I can almost guarantee that this left the wrong impression. The fact that she's trying to avoid you and hide from you means that she's scared of what's going to happen if you catch her. She knows that after going on the bed, something is going to happen that she doesn't like, not that going on the bed itself is bad. There's a common misconception that dogs and cats can know when they're doing something bad or good, when they can't really, but they can associate the events that come after their actions.
That's the idea behind positive re-enforcement, is that it builds the connections between an action, and receiving a positive reward afterwards. Again, this doesn't mean that they know what they're doing is good, they just know that good things happen if they do it.
Dogs are awfully eager to please when they know they're going to get a rewards for obeying, but unfortunately cats haven't been selectively bred for obedience as dogs have. People want cats to be independant. But all that just means that it takes more work and patience to train them, not that it's impossible.
First off, I think that there are two things going on here.
She is using the same spot because it smells like her urine. Cat urine is very high in ammonia so it's extremely difficult to clean out completely. Also consider that even though you might not be able to smell it, your cat still could. Cat's are hardwired to detect cat urine, since they use it to send messages for things like territory borders and when they're in heat.
Stressed cats typically avoid litter boxes. There are many reasons why that is, and I'm not well versed enough in cat behaviour to say why, but for whatever reason, cats tend to go outside the litter box when they're under stress. Considering how she's acting scared for a few days after, I think it's safe to say there's some stress being cause of her fearing whatever punishment she receives for going on the bed. But, it's important to note that it could also be stress from a medical issue, like say a urinary tract infection, and you would want to take her to a vet to make sure.
My advice to you is to make sure that the cat urine is cleaned out of the sheets and mattress completely. Use a strong pet cleaner like nature's miracle to attack the mattress. Luckily we have another question that's been asked about removing urine from a mattress: How can I get old cat urine smell out of a mattress?
It might also be a good idea, if it's a possibility, to restrict your cat's access to the bedroom. Keep the door closed for a week or two and see if she can't get back into the habit of using the litterbox.
The absolutely most difficult part of it (believe me, I've been through it too) is when you find a mess, you have to completely ignore her. You're going to have to act like nothing happened so that she realizes that whatever she's afraid of happening after she goes isn't anything she needs to be stressed about. It's going to try your patience, because it will take quite a bit of time for this to happen, but after she stops being stressed she should be more comfortable using her litterbox again.
I assume you've had her for a while, so she should know where her litterbox is located, but it might not hurt to re-acquaint her with it. If your bedroom is closer to where she normally spends her time than her litterbox, than maybe she's having trouble getting to the litterbox in time. This could be a sign of a medical issue, and again you'll want to have her looked at by a vet.
If it's not a medical issue, you can treat it the same as you would with a kitten. You want her to relearn how long it takes for her to get to the litterbox from different areas of the house. So say the litterbox is in the basement, you would want her to stay in the basement for a week or two, and once she learns how long it takes to get to the litterbox from a room or two away, then you can let her go a bit farther until she knows how soon she needs to make her way to the litterbox from the farthest point in the house to make it on time.
It might not be a bad idea to have a second litterbox at the other side of the house that the other litterbox is at though. Depending on how much space you're dealing with of course. It might not be necessary in a small apartment.
You might also want to consider trying out some different types of cat litter. Cats are pretty picky and like kids and vegetables their preferences can change. Perhaps the litter doesn't mask the smell of their urine/feces enough, or even the opposite where the scent of the litter is too strong.
Finally, something that might help in the training, and calming down stress department. There are a variety of cat calming aids available. Something like the ComfortZone diffuser, assuming it works in helping control the stress and anxiety your cat might be having, would be useful to use if you can identify when your cat is acting most stressed. Separation anxiety for an example, you could turn it on as you leave, and slowly dial it back as you work on training to combat the anxiety.
