Hi - he bit because he was scared and also warning you off. It does sound like maybe doing a bit of taming as Lizzy suggests would help. Our hamster couldn't cope with being handled by more than one person at first when he was first tame, and it set back taming a bit.
It's difficult with children - you have to constantly supervise and show them how to hold the hamster gently and get used to them. Trying to make him go on his wheel repeatedly would just stress the hamster.
He's probably just a bit nervous now if it hurt when your daughter handled him so bit you because he thought he was going to be hurt again.
Sometimes it's all a bit much the first few days and then there's a set back. I would do what Lizzy suggested - start again giving him 2 or 3 days to himself, then start feeding him treats through the bars and talking to him gently.
Our little boy was 6 when we got our hamster and I had to constantly watch him when holding Charlie as sometimes he would try and hold on if the hamster tried to run off, or would hold him a bit awkwardly. It certainly teaches children patience! And also that the hamster isn't a toy and needs gentle handling.
If he was squeaking he could have been a bit scared. I've been there - our hamster used to run at my hand if I put it in the cage before he was used to us! Frightened me a couple of times. Part of that was that he had a bit of cage rage as his cage was too small and totally unsuitable.
I would suggest not handling him while he is in his cage. They are more territorial in the cage. And only handling him out of the cage in a safe area like a playpen or the dry bathtub. I did a little video about how we got our hamster used to touch and being held if it helps - it doesn't show much but has a few tips (mostly picked up on here).
We would get our hamster out in a tube - ie put the tube in the cage with the far end in the hamster ball, and let him walk into the tube and then into the ball, then pop the lid on the ball and gently carry it up to the bathtub.
https://www.youtube.com/watch?v=kjyEme2xcq4