How to finish a quest after the NPC you need to deliver it to, has become hostile to you

You've likely hit a bug.

While you've agreed to help Sinding, it doesn't lock in your choice: if you attack him at any time before the end of the quest (when the curse is removed from the ring, allowing you to take it off), you lock in your choice to help Hircine.

However, doing so is supposed to go down differently: you act as one of the hunters and chase Hircine through the grotto, ending with you skinning him. He's not supposed to appear in jail.

You can try that: kill him in jail and skin him. If that part isn't bugged, Hircine should appear and complete the quest for you. As a reward, the ring will disappear.

Otherwise, welcome to a Bethesda game, where crippling bugs like this are par for the course in the first couple of weeks after launch.

Okay, in my fifth run through Calcelmos lab I finally found the reason.

Just in the second room (you enter the lab, go a bit further, turn right and go further) there is one Wizard Guard and, sadly, a normal Markarth City Guard. When it locates you, it places a bounty worth 5 on you (probably because you shouldn't be there).

When you talk to him and try to pay him to strike out the bounty you will be teleported out of the lab. Luckily the bounty is so small that you can convince him it's too much work to cope with you and he just leaves you alone.

After not killing this guard, I could have the bounty removed on any other city guard after leaving the lab and the other city guards won't be hostile.