How tall is she, what does she weigh, does she bark? Face looks a little like Basenji, body is not square enough, but maybe......a DNA test would give you a better idea.
The dog on the left is Ginger, she's had a DNA test and is 75% Basenji. The dog on the right is Bailey, he's Ginger's son. We believe the father was a Heeler.
Looking at your dog, I definitely see Basenji in her face and ears.
Looks a lot like my Jack. I haven’t done a DNA test because I don’t think it makes any difference. Jack is a rescue and I was told they suspected he was a Rat Terrier mix. My vet, too, says he sees some Basenji in him. He does bark but only when he feels threatened He gets my attention by making weird noises. He also grooms himself like a cat! He is very agile!!
I see Basenji in face, ears, and body shape (especially chest) - beautiful dog!
Don't worry too much about the barking. Some do and some will remain silent for their entire lives. Barking is a continuous series of barks, and most Bs will do just one, without the continuing series of barks - like a lot of lonely dogs will do. He looks like he maybe a B, but you would need to get closer to his bloodlines, but does it matter? He looks like a good dog, and should be a keeper regardless of his pedigree.
To me, definitely a basenji mix. I have seen registered basenjis who are not square, 'curvy' topline. She is not as refined as a female basenji but that might be the mix showing up.
The howling noises are very basenji to me.
A few basenji characteristics she may/may not have:
Licks herself when wet, like a cat, does not have a doggy odor when wet, picks a particular time of day, and goes nuts, consistent with the time of day. Has a hard time controlling herself when a small critter, especially a rabbit is near.
Thinking she is part basenji will buy her a few 'oh well, that's the basenji part showing up' when she does something that a regular dog will get reprimanded for. With Lexi, roll up a newspaper and swat YOURSELF for not giving her a baseni proof environment!
Your life with Lexi will never be boring!
Thank you all! She is amazing and you are all correct, life with her is never dull. With all of the new information that I have now and understanding she may be a Basenji mix, I just hired a trainer (for us it sounds like).
Your dog looks a lot like mine. Our vet says Basenji terrier mix. 
@mvdperez - Not all Basenjis have a freckled belly. I have been in the breed for 25+ years and had a number of Basenjis with no freckled belly. That said in the pictures of her posted you can see the mottled skin under the white hair....
