It sounds like your cat is unhealthy and is attracting the fleas. The cat has low vibrational energy and something needs to be done. It sounds like a health hazard. A sick animal will not be able to resist them and they will have food and breeding ground. If there were no animals to feed on, the fleas would go away.
Is it humid in your area? The fleas will not dry out if it is. The dehumidifier idea sounds like it would work together with everything else. If no dehumidifier but there is a wood stove, dry it out once a week - that will halp too. I have not had fleas for 20 years and just now have them for the first time since then. I used to use salt on the floor (left overnight) and then vacuumed floors and furniture daily. Make sure there is salt in the vacuum bag to dry them out there. The animals have to be washed with Bronner's or Sal Suds with essential oils added. I used to use a phenomenal flea powder made in Australia with eucalyptus and tea tree oil. Not sold any more. Tea tree oil is toxic to cats so be careful there.
Keep all sick animals out of your living space and see if the owners can get them some help. I would move if the people are not doing anything about it.
I know DE works for many things but I am a little wary of using it. I would use less, not more. I haven't had to live with chronic fleas, just be diligent and they should be gone in a few weeks. The cold weather will take care of them outside.
Garlic works to repel them too - I use garlic powder from the grocery store, 99 cents for a large container. Good stuff for all animals and people too. Give daily doses to your animals during an infestation and every now and then for a few days as a preventative.
Apple cider vinegar is good for the skin and is supposed to repel them. We put it into their drinking water from time to time. It's especially good for hot spots on dogs, heals skin rapidly and reduces itching. Try a rinse or a spray for the pets and the house, try adding essential oils as seem appropriate. You will have to judge if it's too humid and if more moisture would just cause more problems.
Homeopathy can help for the pets too - sometimes I learn a lot from what's going on with the critters! If it's not a good situation, if the energy of the whole place is low, it will wear on your health and your family too. Take care.