I had been playing through this quest and accused the court wizard of the murders in Windhelm. I went to see him in prison and he told me to look for the real killer at night. I ran frantically around the city but couldn't catch up to the arrow indicator on the map. I did find a dead body though and no one seems to want to acknowledge that it is there. Everyone just walks past it. So, I went to my house and found the real murderer inside, trying to find his bloody altar. I can hit him until he falls to his knees but he gets back up and carries on. If I try to hit him more after he goes down, guards come into my house and start attacking me! I have no conversation options with the Jarl or his steward to tell them the real killer is in my house. How can I complete this quest?
I don't know if there are more ways to complete this, but I waited in the area until the right time and like you couldn't find anything.
I then walked right outside the area and heard a scream and a new arrow popped up leading me to the market.
If you're fast enough, you should be able to find the murderer there trying to kill a woman and you'll be able to stop him.
From the UESP wiki's "Blood on the Ice - Bugs" article:
If playing on the PC, make sure to install the Unofficial Skyrim Patch (USKP) as it fixes the bugs left unfixed in the 'Blood on the Ice' quest by the official patches. List of bug fixes by USKP related to the Blood on the Ice quest here.