How can I complete the Blood on the Ice quest?

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?

From the UESP wiki's "Blood on the Ice - Bugs" article:

Wuunferth Arrested Variation

If you speak to Wuunferth directly, instead of Jorleif, then Calixto will attempt to murder Arivanya in the Stone Quarter. If you choose to have Wuunferth arrested, then Arivanya will be found murdered. For the final showdown, a random, female NPC will be chosen as the market victim. If the game runs out of potential victims, one of two bugs will trigger:

• After speaking to Wuunferth, the journal will immediately update to stage 130. After chasing Calixto into Hjerim, he cannot be killed. He then will not leave Hjerim and becomes hostile when you enter.
  • This bug is fixed by version 1.4 of the Official Skyrim Patch.
  • (PC) To complete quest, exit Hjerim and use console commands: prid 1B11D and moveto player.
• Calixto will stay inside his House of Curiosities and will never attempt his last kill.
• If you buy Hjerim and arrest Wuunferth, when you get the objective to patrol Windhelm's streets Calixto may appear inside Hjerim, in the room behind the wardrobe, and use any equipment you may have left in it. He will be immediately aggressive and killing him advances the quest normally.

I can hit him until he falls to his knees but he gets back up and carries on.

Updating Skyrim should fix this bug. In March 2012 (PC) and April 2012 (consoles), update patch version 1.5 was released. One of its bug fixes is:

Fixed issue where Calixto would fail to die properly in Blood on the Ice.

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.

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.