The Darkrai player sends up first, since a lot of time passes between the Lando KO and the Darkrai KO.
~Landorus uses Land's Judgment to KO Darkrai EX. Darkrai is discarded.
~Lando player takes 2 Prize Cards. The Darkrai player replaces Active.
Between turns.
~Landorus is given damage counters from Poison, and is KO'd as a result. Landorus is discarded.
~Darkrai player takes 2 Prize Cards. The Lando player replaces Active.
It is now the Darkrai player's turn. Play continues as normal.
A KO in between turns is a lot different than a KO during a turn. For example...if the scenario you just mentioned happened after Time was called in a tournament match, and the Lando attacked on Turn 3 after Time was called, you would count prizes before the turn ended - and if a winner was declared there, Lando would not be KO'd from poison.