People DO NOT RSVP. They just don't do that anymore.
If you aren't friends with these people then why did you invite them to your home? Why don't you make friends with them and then you can just pick up the phone and call them to talk about the party while you're planning it.
They'll have already said if they could come or not. Having strangers come to your home is just...I just don't do it and expect them to call me and say they're coming. I call them and talk to them. Then the ones that said they could come or might come are the ones that get the invitations.
I'd say call each and every parent and let them know the party is cancelled since no one could make it. If they say "We were planning on coming" you can let them know that you didn't know that and you're sorry but you've already cancelled everything.
I think you can have the one that RSVP'd over to hang out, you're not that sick and it won't be a birthday party per say, just have a play date.