How can I affix this wood transition piece?
In view of the new information, I would remove the mortar in the gap.  Seal the edge of the wood floor at the sub floor with a waterproof caulk  and use a funnel and self leveling (cement based) underlayment to make a flat base. Tape both sides of the gap with some blue tape to make any cement spillage an easy cleanup. When setup (24-48 hr) and well cured, attach the transition to the gap, I prefer the polyurethane adhesives (PL) over LN.  (It is stronger, stickier and more flexible, over time).  

As noted previously by Tester101 in chat, weight the transition in place overnight.