Can 2013 MacBook Air output to 2560x1080?
I believe the problem is your HDMI cable - it's not capable of displaying a resolution of 2560x1080.  In other words, it's pre-HDMI 1.4 specification.  (Also, the link provided to the cable in the original question is out of date; it's no longer available).

Per Apple Support, the 2013 MacBook Air (11" and 13" model) has a Thunderbolt port which also supports mini Display Port.



Since the HDMI cable referenced in the original question is 4K capable (HDMI v1.4 specification), the cable is not the issue.  What wasn't mentioned was the HDMI adapter. I believe this component to be the "weak link" in this chain because if the mDP to HDMI adapter isn't 4K capable, nothing else will matter.

I recommend getting an active mDP to HDMI adapter (rather than passive) and then connecting your HDMI (v1.4 or above) cable from the adapter to the TV.   



That said, I would only recommend this solution for this particular setup, because the LG monitor 29UM57-P only supports HDMI connections.

Ideally, you would want to use mDP to either mDP or DP connections rather than convert to HDMI, DVI, etc., to get the most reliable connection(s).  Less conversions means less points of potential failure.