So our Yorkie is almost 6 months old, and she is still having a hard time with potty training. I know we can't expect perfection yet, but I would like to see some progress!

She is fine when she is in her crate at night, but, when we let her out during the day, she'll pee on the floor--even after we've just taken her outside and she's done her business outside. I've tried washing the floor, but I think the scent of her urine is now in the hardwood, and she thinks it's okay to pee (and poop) in the house.
I've tried catching her in the act and moving her outside, but it doesn't seem to work. She knows what she is doing because when I catch her, she hangs her head like she knows she's done a bad thing.
I've read about crate training during the day-only letting her out to go to the bathroom outside and play for a bit and then put her back in and repeat later. Have any of you tried this method? I'm feeling bad about leaving her in the crate so much, but she is peeing everywhere in my downstairs. Any suggestions? Is crate training a humane way to train given the fact that she'll be stuck in that little crate so much of the day and night? Thanks for all your help. She's a cutie, but I'm beginning to wonder if we'll ever move forward in the potty training department.