Kelly, I agree totally with the posters who are saying YOU have to discipline and parent your daughter. If she won't listen to a word he says, but you don't back him up on anything, and don't do any disciplining, then of course she's going to hate him - he's doing the parenting, the difficult bits of parenting, while you standing by and not supporting him. From her point of view, he isn't her parent - and you are reinforcing that message by essentially ignoring him too. She's learnt it's ok to ignore him because you've given her that message. When she's rude to him, you've got to pick her up on it!
Maybe you think he's too strict. If that's the case, then you and he need to sit down and agree some boundaries, what battles you will choose to fight. And then BOTH of you need to back eachother up. You are undermining him totally at the moment. If I were him, I wouldn't put up with it, I expect he's incredibly frustrated. You're making him be the bad guy all the time. He isn't able to have any fun with your daughter because you've allowed her to disrespect him. You said they enjoyed cycling together in the summer - so when she was able to have some fun with him, their relationship improved. Not really a surprise is it?!
And your daughter must be very angry - she must see a mum who lets her do what she wants (which includes being rude to the other adult in the house) and her mum's husband, who isn't her dad, as the only person who picks her up on her behaviour. No wonder she is angry. This may sound harsh, but you have allowed that anger to develop by not acting as your daughter's parent.
My DP and I don't always agree on parenting, but we back eachother up as a team in front of the DCs (we both have DCs from previous marriages, and none together, so we are both step parents).
I think you need a serious talk with your DH - to agree your "rules" and boundaries. And a serious talk with your daughter, where you explain those rules and boundaries, and that you will BOTH be enforcing them. Your daughter probably won't like it at first I'm sure. But I expect you'll all be happier in the long run.