Is it possible to go-around at Lukla Airport?
No, it is almost impossible.

The only runway (06/24) Lukla Airport is just 1500 feet long. This article lists some interesting facts:


  The topography of the place makes any go-around impossible. At the
  south, the runway is the end of an angled drop of about 2000 feet.
  This cliff is fenced off as a precautionary measure. At the northern
  end of the runway there is a huge mountain terrain. It leaves no room
  for error.


A pilot mentions the same here:


  "Because there is no way to go around again, we have to calculate many
  things like air speed, tail wind, fog. If you don't do the proper
  calculation or proper exercise, then it (an accident) happens."


Planes land on runway 06 (Heading 060 or ENE) and takeoff from runway 24 (Heading 240 or WSW). This picture shows the cliff on North:



There have been several accidents at this airport.