Unfortunately I don't have definite literature that will give you an answer either way, but I can tell you quite confidently that, safe or not, the borax won't completely get rid of your flea problem. The safest way to go would be to go to your vet and get some flea meds for your kitty. Some of these are quite expensive, and they cover a range of internal and external parasites. My animals are all on a product called Revolution (selamectin). Not only does it get rid of ticks and fleas, but it is also a preventative treatment for fleas (adults and eggs), sarcoptic mites, ear mites, heartworm and many intestinal parasites. It's also a very safe product to use.
If you're looking for something that's a tad less expensive but still very effective, I would go for Advantage. It's a few drops of liquid that are applied monthly to the back of the animal's neck (VERY easy to do!). Most vets do require the animal come in, but usually it's just to get an accurate weight (which doesn't cost anything). Once the animals are on Advantage used every month, then your flea problem should virtually disappear! Fleas live on animals, so without animals to live on, they won't be in your house.
Both Advantage and Revolution are very safe. I'd keep your child away from the cat for a few hours after you apply the medication, just in case. Maybe you could apply it when your child goes to bed.
Anywho, I hope that was helpful! Good luck in achieving a blissfully flea-free home! :)
3 years as a veterinary assistant, and many years as a pet owner