It's your house that has fleas, as well as your cat. The cat is getting them from the house. New generations of fleas hatch out every 2 weeks, and the eggs also lie dormant for that long. They are very much self-perpetuating as long as there is ANYTHING in the home they can feed on, and that includes people (fleas that infest people are a different species, but cat fleas are not above snacking on us occasionally to survive.) Have the vet give you a flea repellent drops treatment for your kitty, and visit the hardware store for flea bombs for your home; enough to bomb it 3 separate times, because the egg-casings are resistant to the fogger mist poison. You have to actively wait out the hatching of 3 generations of fleas and be sure to poison all of them.
As to what to do, the answer is simple. It doesn't cost a lot of money but does require intense effort, and it takes a total of 7 weeks to do. Here is the routine: The carpet / rugs and all upholstered furniture all must be double-vacuumed with the dust waste removed outside the house, then repeat that again every week for 7 weeks. All "soft" separate goods made of fabric that cannot be washed and that humans handle regularly must be sealed in black trash bags for that long. Everything else will be treated by flea bomb. You use the trash-bag treatment during fogging for all eating/cooking utensils and any human- or pet- centric soft goods that cannot be washed so that they are not touched directly by the mist, and also all non-refrigerated foodstuffs, and then you set off the foggers in your closed-up home according to directions, taking yourselves and your pets out of the house for a few hours. Then you take all your dry food back out, wash your dishes and all the bedding, curtains, and clothing, and wipe down fixtures and hard furniture. Then wait 2 weeks and repeat the necessary steps to do the fogger again, then wait 2 weeks more and do it a 3rd time. Then you can liberate the unwashable soft goods from their trash-bag prisons, because your fleas should finally be eliminated at that point; the 3 generations sealed in without air will have died as well as the ones poisoned by the foggers.
If you find any traces of mice be sure to seal any places they can get in, and also make sure to put out cat-proof poison dispensers for them if your cat is not a good mouser.