Giving Chronicles of Nchuleft to Urag – ‘Fetch me that book’ quest

It looks like you're suffering a bug.

According to uesp

If the Markarth Guards do not warn you off after having investigated the Inn for Eltrys and you get stuck (Guards won't arrest you) when returning to him after finding out all you can about the Conspiracy, if you wander the streets and do meet the Guard who warns you off and then returning to the Temple, you will still get arrested (or attempted to get arrested).

So it looks like you need to talk to all the guards in the city.

Another possibility is this (I hope this isn't your problem - no way out):

If you enter The Warrens before your first talk to Eltrys in Shrine of Talos, a Nord named Dryston will approach you. He will tell you that "it's time you learned a lesson" and brawl with you. Once you beat him, he will NOT tell you that "N" is Nepos the Nose as you did not get this quest yet, this will lead to a BUG where you can not complete this quest 'find out who "N" is' once you read Weylin's Note (if you read it right after you find it by lock picking the Weylin's door, it will not start the quest, as you did not speak to Eltrys yet).

Very odd - I found the wedding ring on the witch myself, so it does sound like a glitch. Unfortunately as an Xbox user you can't abuse the console into solving the problem.

If I remember correctly if your speech is high enough, you should be able to persuade/intimidate Ysolda into telling you the next location for the quest. This looks to be your only option if you don't want to pay.