The vet can supply a liquid flea killer to be rubbed in if the cat will tolerate it. Although I use Frontline spot-on monthly, one of mine once came home from a week-long absence with large fleas lined up around her ears, which I had to pick off with a tick remover. For the house the best option may be room fogger/bombs, to be done an area at a time, moving around the house.
I shall never forget as a child coming home from holiday to find my faux fur bedside rug absolutely jumping with cat fleas! My vet neighbour says a poorly or elderly cat has less resistance which could lead to infestation despite preventative measures.
Good luck!