I'm pretty sure I accidentally killed Arivanya early in the game and now I am unable to start the Blood on the Ice quest. There have been many fixes for this quest for various bugs but not this one. I'm sure others have this problem and would very much appreciate it if the issue was resolved.

Well, if you accidentally kill any of the NPC's that are crucial in finishing a quest line it is not a bug.

But, do you know for a fact that you yourself accidentally killed her or did you just find her dead? because finding certain NPC's dead for no apparent reason that are crucial to 'Blood on the Ice' is a bug. I once found one of them dead in her home lying in her bed, I know I didn't do it because I had never even been inside of that house.

But, if you accidentally kill one or find one dead from an obvious dragon or vamp attack that it is not a bug that can be fixed.

I have not had a problem with that quest in a long time though, not since before DG.

A rule of thumb for rpg's: Do not kill a named NPC unless (1) you're attacked or (2) you're directed to by a quest. If you should kill one accidentally either stop and revert to an earlier save, or soldier on and hope they weren't anyone that's quest related.

This is especially important for console gamers, but it makes things a lot easier for PC gamers too.

Unfortunately Arivanya is needed for this quest to trigger, and there is a workaround but only for PC users.

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But the quest have 2 different workarounds:

If the Dragonborn accuses Wuunferth The Unliving to be the true identity of the "The Butcher", and he is locked away, Arivanya will be the next victim of "the Butcher" in the coming days.

However,

*** If the Dragonborn chooses to cooperate with Wuunferth and tracks "the Butcher" to his next crime scene

it's possible that Arivanya will appear as the next victim in which the Dragonborn may save her in the marketplace of Windhelm by killing the butcher before he stabs her.

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So, if you have killed her accidentally, go back to a previous save and choose the second *** workaround !!!!!

GOOD LUCK !!!!!

Like Mark mentioned you will have to reload an earlier save prior to killing her to fix this issue.

Also:

This does not directly relate to the original issue, but If you cooperate with Wuunferth (instead of turning him into the steward), the chest near the master bed in Hjerim will never appear later in the game when you purchase the home. I have personally tested this out myself and it is also listed as confirmed on the wiki. This bug may not seem significant enough to many people to worry about, but I thought I should point out that there is a direct connection between your choice during Blood on the Ice and whether or not the above mentioned container will appear in your home later in the game. It seems like a weird reason for the no chest bug to trigger, I know, but I can assure you it is the cause.