She definitely needs to see a vet. My cat was peeing outside the litter box on occasion (usually once or twice a month). I finally took her in to the vet and found out that she has cystitis (inflammation of the bladder). She also had a small amount of blood in her urine. Even though there doesn't seem to be a pattern, your kitty may have a medical issue. Is there any way you can have an older friend or family member take you and your kitty to see the vet? Also, I would recommend getting a new litterbox. Cats are very good at hiding their pain, but they will associate the litterbox with their discomfort during urination. Basically, she may be blaming the litterbox for her pain and therefore avoiding it on occasion.
If that doesn't work, you might try reintroducing her to the house. Start by keeping her in the room with the litterbox, and slowly let her expand into other areas.
Last but not least, your vet can be a good resource, even over the phone. Call and explain the situation, and ask if they have any other suggestions.