Dawn baths will kill the fleas on them at that moment BUT don't bathe them right before or after applying topical flea medication because you'll just wash the meds off.
You'll need to vacuum daily. Yes, even hardwood floors. Throw out the bag/dump out the canister each time. Wash all bedding the cat likes to sit on. And keep up on a flea preventative. Frontline, while a good product, does unfortunately seem to have resistance problems. I do second using Revolution, or Bravecto (which is a three-month application BUT also basically as expensive as 3 months of Revolution, but it may be worth it if you don't want to apply stuff monthly!). You can use something like Vet-Kem spray in the house to kill the fleas in there. Outdoors you'll probably want to hire a pest control company.
It's going to take at least three months to get rid of an infestation, unfortunately. There is no substance that will kill all the flea life stages so you have to keep up with preventative measures as long as it takes all the fleas present to go from egg to all the larval stages to pupa to adult. That's why you'll seem to get the problem under control, and then it pops up again--you just killed the current generation of fleas but there's more waiting to hatch!
Yes, it's obnoxious. Don't panic, just keep at it, and it'll go away eventually.
Keep an eye out for tapeworms on the cat too, those are carried by fleas and it's not uncommon for cats to end up with tapeworms after a heavy flea infestation. Again, don't panic, it can be treated, just be aware (and no you can't catch tapeworms from your cat).