The first thing that you must do is take Dcat to the vet and get Bravecto or a similar prescription flea treatment. Frontline has stopped working in many parts of the country due to the fleas developing resistance to it and imidocloprid preparations are also losing effectiveness. Vets will only prescribe flea treatments if they have recently seen Dcat and he is "under their treatment".
A quick overview of the flea lifecycle: like butterflies, they have an egg phase, a larval phase, they pupate, and then they hatch out into adults. If you can break that lifecycle consistently at at least one stage, you are golden.
The adults can only lay eggs after a blood meal taken from Dcat. The Bravecto turns Dcat's blood into flea poison, making their first meal their last and stopping them from laying eggs. So once Dcat has Bravecto on him, let him go everywhere in the house so that adults will hatch out, jump onto him, bite once, and die. This breaks the egg-laying phase.
However, you still have existing eggs, larvae, and pupae in your home. The pupae are really sticky and won't come off your carpet, but the eggs and larvae will go up in the vacuum cleaner so vacuum throughly. If you want to use flea foggers to kill eggs and larvae, shut Dcat and any other pets out of the room that you are fogging and keep him out for at least 24 hours. Foggers rarely kill pupae because the poison doesn't penetrate the casing, but the poison will stay in your carpets and kill adults as they emerge. If you buy a fogger that contains s-methoprene, that will also stop the larvae from pupating, breaking the flea lifecycle at a second stage. As long as you don't use a carpet shampooer, the flea killer will stay in your carpet for up to six months.
If you have a carpetted kitchen and wish to apply flea killer to the kitchen floor, use a spray can low to the floor and put dust sheets on the worktops. Insecticides are harmful to humans and pets if inhaled or eaten, so spray quickly and leave. Don't use a fogger in the kitchen.