No pets, fleas in home…new cat arrives in the next month!!!

Hi,
 
Thanks for reading my question! Recently I went on a work trip to Poland and when I arrived home I was covered in bites.
Initially I thought it was bed bugs but couldn’t find any evidence, finally a month on I found a flea! Since the first flurry of bites 1.5 months ago, I get maybe 1 a week. I have caught a total of 5 fleas, 1 on my sofa and 4 in the bed (I think they were on my PJ’s which I also wear whilst on the sofa). 
 
We had pest control round who found that we have a rat in the basement that was coming into the flat at night (shudder), we have now blocked up holes and pest control believes the rat has taken the bait so likely died. So my thinking is that the fleas were falling off the rat when it came into the flat…the worst. 
 
My question! It’s been six weeks now, I most recently found a flea yesterday on the sofa…Our plan this weekend is to do a huge clean, vacuum and mop EVERYTHING then put some spray on the sofa/carpet/floorboards. My big worry is that my new cat is arriving in the next month, we don’t have an exact date, it could be any time….I am terrified once she arrives, any remaining eggs with hatch and fleas will latch onto her and we will be back to square one. 
 
Is this a likely possibility? If so what would you suggest I do? I can try to delay her arrival if absolutely necessary…
 
Thanks 🙁
 
Rosie