This is what I find works.
Use a good quality house flea spray.
Wash all material that the animals have contact with. Cushions, throws, bedding etc. Eggs hatch (I think) about every 3 days. So the washing is done every 3 days. Do it all at once, so the cat is not moving eggs from one place to another.
I do not bathe the cat, but I do use a mild solution of white vinegar and bicarb (which I read dehydrates the fleas) and baby shampoo in warm water. Using a comb regularly dipped into the water solution, the coat gets wetter, the fleas cannot jump around in the wet hair and the fleas eventually end up in the water. When it is very bad, do this once a day and a regular brushing once a day and then go down to once a day with the solution with no additional brushing and gradually reduce further. Do not stop until there are no fleas for several days. Eggs can be harder to get rid of. The benefit of this was the cat's coat was the glossiest and smoothest its ever been and the cat smelled lovely too.
The vibrations of the vacuum cleaner make the eggs hatch apparently. Some people swear by regular vacuuming. The opposite works for me. I vacuum thoroughly once to pick up eggs and fleas (and flea spray and wrap the waste from the vacuum) and do not vacuum again until it is under control. In between I use a sweeping brush or dustpan and brush.
When it is under control then I start vacuuming again. You will know when this is because any fleas on the cat are very small (young) and not all over the place and gradually they disappear altogether.
I brush the cat every day now. I also use a great brush called a 'Zoom Groom' which really gets to the under layers of hairs, so removes some flea hiding places.
The cat has a permanant flea collar now which I change every few months.
There is no quick process that I know of but having had to deal with this issue several times (but not recently) until I found a solution that worked for me, I know how nasty this issue is and hope that this post helps. Good luck!