Lay down with kittens, wake up with fleas

The vet combed the kitten and couldn't see any fleas. She warned me they might be in the egg stage. I have a feeling they are not eggs any longer. I still can't see them, but something has me all bitten up! Quick search on Google tells me Piper is too young to use chemicals on and that I will be washing every bit of bedding AGAIN. The treatment is to bathe him with Dawn dishwashing detergent. Thank goodness he is tiny and my sink is deep. I think I can do it by myself. 

We were lucky that our daughters never caught lice. Dumb luck since elementary schools are the place even the cleanest of kids get them. There was a time when my nephews and niece got lice and kept getting re-infested. Their mother had to wash all the bedding and all their clothes time and time again. What a job! 

Right now he is running across the top of the couch and ducking down under the pillows in the corners. He is as cute as cute can be, fleas and all. 

Dear Readers, if you have hints on how to deal with this pesky problem on a tiny kitty and a carpeted apartment, please let me know. Thanks!