Have you phoned the council? Some councils will fumigate for you (you may have to pay / move out for 24 hours but probably cheaper than buying the flea stuff yourself!)
I got a really nasty flea infestation a few years ago and unfortunately treating the cats alone won't do it.
Here's what I did:
Washed everything that I could at high heat in the laundrette, then tumble dried, then binned anything that had shrunk beyond repair. This included all of my clothes, all bed linen, all curtains, sofa covers, mattress cover, pillows etc in the entire house including stuff that was packed / stored away. Whilst unpacking linen cupboards, wardrobes etc I also
Hoovered them and then sprayed them, left flea powder on the floors.
I also hoovered twice a day (as much as I could!) for two-three weeks with a handful of flea powder in the Hoover bag.
Sprayed topical flea killer on all furniture.
Flea powder under sofa cushions, under bed etc.
Bought flea smoke bombs (one per room) and used these twice (two weeks apart)
Frontline combo on the cat plus combing plus topical flea killer spray every day until there were no more coming off her.
It was a huge amount of work and chemicals, it cost me well over £100 in the powders etc. as you have a small baby I would DEFINITELY look into having it done professionally, and moving out whilst it is done.
BUT remember that the fleas travel! Take as little as possible with you and make sure all clothes etc are freshly washed on a high heat (60+) and suitcase, buggy etc thoroughly cleaned, hoovered and sprayed (air them outside afterwards) to minimise the chances of you taking them with you!
Honestly it is really shitty, I feel for you. Luckily they seem to prefer some people to others - if you have hundreds of bites and your baby none hopefully that means they are not "tasty" to fleas! I didn't get bitten at all even when my carpet was seething with the buggers, my friend looked like she had measles after ten minutes on the sofa. 