Find the Quill of Gemination under Lake Honrich in Skyrim

Check to see if he isn't sitting on a chair in his quarters. I just spent 15 minutes looking for him but could hear him coughing.

As you're playing the Legendary Edition you have all of the DLC, including Dragonborn. As such, you should be able to find Gunmar in one of the following locations:

In the Skyrim overworld:

• Fallowstone Cave
• Brood Cavern
• Crystaldrift Cave
• Bonechill Passage
• Clearspring Tarn
• Cronvangr Cave
• Honeystrand Cave
• Pinepeak Cavern

Or in the Solstheim overworld:

• Northshore Landing
• Horker Island

(source: TES Wiki's Gunmar page)

If you can't find a map marker for him in the Skyrim overworld, he's probably at one of the two locations in Solstheim.