Where do the deer bed in your area?
Are there any areas with thick brush?
Were you in a tree stand shooting down or on the ground?
A deer hit from above with no exit wound coming out the bottom will bleed internally and will leave very little blood trail if any at all.
Chances are if you missed, you would find the arrow if in a tree stand.
I understand te going in circles but "d reccomend going in the direction he ran. How far did you hear it run after the shot?
That "mule kick" has me wondering.
I've tracked a lot of deer and have seen lots of stuff. Found a few others gave up on.
Tracked many without recovering too.
If you can pull together some friends (wife) spread out and search for a body. The meat may be spoiled by now but maybe you can salvage the cape for mounting if it has been cool in your area.