I am a life long animal lover, and I try to use the most holistic treatment possible where my own kitties are concerned. My sister manages a busy veterinary clinic, and my niece is a veterinarian. Since I have adopted two stray kitties who were infested, we've gone over the pros and cons of all the options.
With fleas, the safest way to get rid of them for sure is to stop the reproductive cycle in its tracks.
I recommend taking the cats to their vet for one of the topical flea treatments like Frontline or Advantage. It's important that the cats be seen by the vet first. These products are not without side effects, but if the cat is healthy, they should be fine to use.
When you talk to your dad's vet, find out if his kitty has been seen in the past 6 months. If he (she) has been, the vet might be willing to sell you the treatment without seeing kitty for "well kitty" check up first. Most vets will be willing to skip the check-up if the animal has been seen in the lat 6 months. The other two cats are yours, right? They will need to be treated too. Once you discover fleas, all animals in the house will need to be treated.
DO NOT TREAT THE HOUSE WITH ANYTHING AT ALL! All flea and tick treatments can interact with each other, with deadly consequences. Fogs, powders, sprays, etc., are the most dangerous of all the flea treatment products. You do not want to expose anyone ~ especially children or animals ~ to those chemicals.
When applying any of the topical products, you want to apply it to the skin at the scruff of the neck or between the "shoulders". Make sure it is getting down to the skin.
The way these products work is this: as the fleas bite, they are either killed or rendered infertile, so the reproductive cycle is stopped. In very little time, all fleas in the environment will have died off.
Keep sweeping and cleaning, but change the bag or empty the vacuum often. You can also put moth balls or flakes in the bag and that will help to control the fleas.
This isn't a cheap fix, but it is the best one available, and the safest for the humans in the home. Especially for the children, who put just about anything and everything in their mouths.