Just my 2 cents, there is a TON of work that goes into breeding, even if you aren't showing your dog or doing performance events.
When I fell in love with this breed, I knew that some day, maybe several years down the road, I wanted to breed. I've had basenjis for over 3 years (a drop in the bucket), and the more I learn about the breed and breeding, the more I realize that I don't know anything!!!
Its difficult enough to work with just 1 dog on a daily basis on just regular dog things, like socialization, exercise, feeding, grooming, cuddle time, etc etc etc. Most breeders have more than 1 dog in their breeding program, so multiply that work times the number of dogs you have. Now consider a puppy. I don't know how old yours is, but if its an adult, think back to how much WORK goes into a puppy. Multiply that by 6 (which is an average basenji litter), and add in your adult dogs. You say you don't have enough time to show because you have a lot on your hands raising your daughter by yourself; how on earth will you have time to raise puppies!?!?!? I'm not saying this to be mean AT ALL, I'm just trying to give you an eye opener.
I had absolutely no idea how much work went into a litter, but I am extremely fortunate that YodelDogs has taken me under her wing to be my mentor. I was lucky enough (since I was unemployed at the time), to visit her litter this year at least once a week for the first 12 weeks of their lives. Just going over for an afternoon could be overwhelming with 6 puppies!! CONSTANT cleaning up (even though towards the end they were almost completely litter trained), constant noise noise noise, running, jumping, trying to eat your hair, your earrings, your brand new favorite sweater, the bow on your shoes, anything they could get to. Lots and lots and lots and lots of work. I took one of the puppies home for a few nights on a "socialization loan", and although it was a blast and the puppy was super cute, I wasn't almost willing to pay Robyn to take the puppy back
I just recommend you listen to what Tanza & Lvoss are trying to say; puppies are a LOT of work, and its not something you can fully appreciate until you're knee deep in puppies. Even if you don't have time for showing and performance, find a good, responsible breeder who will mentor you and guide you.