@dagodingo - just a note that Basenji ears are not supposed to be pointed.... the ears are supposed to be hooded.
@salemsockmonkey she came from the ASPCA
It is hard to determine how big/small she is by your pictures... according to AKC standards: females should be ~16" tall and ~22#s (40.64cm and 7.26km). I find the easiest way is to see how tall they are when they are standing along side you and then measure your leg (barefoot from the floor to the spot they reach). I'm old. And odd. 
Now, that said... a pup can look chunky at 22#s, or healthy at 24#s. Everyone carries their weight differently and activity levels vary... it is possible to have a slender, but muscular, dog weigh in heavier than you would expect. Muscle mass is denser and heavier than fat (on humans and dogs). 60#s is huge for a B (so she is almost certainly a B+), but her appearance seems to indicate that her dominant genes are Basenji.
A DNA test would be worthwhile -- especially in this case where there are so many apparent markers for the breed. You are the one that gets to decide how important that is. Please let us know the outcome if you decide to do it! 
It is important to trim their nails to keep their feet compact, - like a cat's foot, and i have found the best way is with a Dremel or similar rotary tool, with sand paper attachment (it could be emery paper attachment). You can grind her nails down without hurting her, instead of trying to get it right with clippers.
@redial She won't let me get anywhere near her feet. She starts becoming really aggressive when people start playing with her nails. I don't know if she was abused with that before I got her, but I can't do them. And I am afraid to take her to a groomer for fear she will bite them. I try to let her walk on asphalt as much as possible, but it is hard when she sees another dog and won't stop barking. So she doesn't get to go on walks much. She has a giant backyard to play in though
A good groomer will know how to deal with a difficult dog, but you should work on getting her to accept having her feet handled. This would be a gradual process, using positive reinforcement. Use food rewards and don't start with her foot if she is very resistant, possibly start high on the leg and work your way down, treating for each closer approach to the feet and stopping whenever she indicates she is getting uncomfortable. Watch for subtle signs, like freezing, before you trigger a more aggressive reaction. Don't expect instant results, work carefully as she begins to associate the handling with something good, the treats. With patience over several short sessions you should be able to desensitize her and eventually work up to actually trimming the nails. Possibly she has had a bad experience with nail trimming, or just was never properly taught as a puppy to accept handling of sensitive areas.
You could also work on her dog aggressiveness/excessive barking to make your walks more pleasant. Have you done any obedience training with her? You can distract her from other dogs by asking for her attention, and when she focuses on you (and stops barking) you can reward her. Another method is to just watch for other dogs and change direction to avoid them, if your purpose is just to give her exercise.
@eeeefarm I've had her for 4 1/2 years. I've worked with her so has my dad (hes train all of our dogs growing up to be the most behaved dog you'd ever met) nothing we've done has helped her aggression. Good thing is she gets along well with other dogs in the house and cats too. Its just when she is on a leash (I've tried getting her attention with food and saying her name) I try changing direction but she tries to pull me back towards the other dog. She also gets aggressive if the dog is on the other side of the fence, unless she's met the dog on the outside with no leash on.
As for the nails I've tried the positive reinforcement, with food and pets and soft kind words and working my way to her nails. Nothing has helped. I might actually need to call in a behavioural specialist to help with it.
She would never actually bite a human even when we are playing. The moment she feels my hand in her mouth she immediately backs off.
A city inspector walked in our back yard the other day and she stood in front of him barking but never attacked.
I hate that when people see her and she starts barking people see her as mean and want to avoid her when she actually would never hurt a fly (literally she leaves the dang things alone)
O. K. I see a couple of things in your post. How much does she weigh? I ask because although Basenjis can be difficult on leash and are strong dogs, they sure can't pull me where I don't want to go! If control is an issue, I find a martingale collar with a wide neck works well. However, it would be much better to find a solution to this problem. Have a look at this [video].https://www.youtube.com/watch?v=HwNe5COoGhk
Different breed, but I was very impressed with how well this guy's approach worked. Could be something worth trying if you have the patience for it.
You say she won't bite you. So what about her feet? Can't you just insist gently that she let you handle them, and reward with treats for cooperation? What does she do, specifically, if you try to hold them? I used to trim my dogs' nails with the dog upside down on my lap, and it worked for all of mine, not that they loved the process, but sometimes you just have to say "yes, I am the boss of you, get over it!"
