Sorry missK I disagree (and I am a vet). At least 90% of a flea infestation is in the environment, it will take months to get rid of them if you only treat the cat. and they are not just noting the cat, are they?
So, go to the vet and get either advantage or advocate spot on for the cat, or comfortis tablets, plus a worming tablet to cover tapeworms, plus a spray for the house that contains permethrin and Smethoprene (acclaim, indorex, Nuvan staykill, RIP flea - all basically the same thing)
Get your DH/DP to do the spraying, go out if it makes you feel better and ventilate the rooms before returning. There is a life stage called a pupa which is pretty invulnerable to any treatment, like a caterpillar in a chrysalis. You want as many as possible of the pupae to hatch before you spray, and they like warmth, humidity and vibration, so heating on, bowl of water on the radiator, Hoover first, then spray. Treat the whole house, concentrate round skirting boards and under heavy furniture that is not normally moved. Get your partner to hoover daily for several days. All bedding on a hot wash but don't spray it.
Continue to treat the cat monthly with one of the products mentioned not with frontline which is useless but you don't need to re-treat the house, the spray lasts a year (only the insect growth regulator Smethoprene which prevents the fleas developing to the next stage in their cycle). With the best will in the world you will have a few pupae survive but when they hatch either the short acting permethrin in the environment, or the treatment on the cat, will kill them off without them surviving to breed.