How old is she?
what is the breed?
Where did she come from? (store, kennel etc.)
At what age did you get her?
What do you do when you find the accident?
Beside taking her out every half hour and the paper towel stuff what have you tried?
Thank you for that information. Since she came from a place you don't know anything about it is possible that they ruined her for potty training. Unbelievable but true , some people still think that if a dog messes in the house then you should put the nose in it and holler at the dog. Dogs do not learn anything from this other than to be afraid to eliminate in front of people.
It has not been long since you have had her that long so it is going to take time to build a trust with her. She may be a more submissive dog than the brother and may frighten easily in certain situations , all depending on how she was previously treated.
What I would do is to continue to crate her to spare your home when she cannot be watched. When you are home with her tether her to you so she must follow you around and you can keep a very close eye on her. If she squats do what you have been doing, however when she is eliminating put a command to it. Something like Go potty and continue saying that even as she is eliminating until she stops. Then praise and give a special treat. It can be a piece of hot dog, or a special dog type treat that she will only get at this time to further encourage her to want to go outside. Outside with you and with the command will begin to equal something good for her.
It may take some time so be patient and try not to get angry in front of her. I would also start doing some positive obedience training with her to further your bond. I am going to direct you to a site for this. The training tools are not expensive, maybe three dollars and whatever treats you want to use.
` Clicker training
video, shows how clicker training is done http://www.clickertrainusa.com/clicker-training-videos.htm