Why didn't my station complete the route?
A Train Station allows its owner to use one, and only one, of the
  routes belonging to another player, into (or out of) that city to help
  him connect the cities on his Destination Tickets.


As per the instructions (emphasis mine), a station can only be used on a single route into or out of the city on which it's placed.  Even if the route from Riga to Kyiv was complete (and I'll assume you double-checked that), the Kyiv station could not be used to complete the ticket if it was already being used to connect a different route (i.e. via Kyiv-Warszawa, Kyiv-Smolensk, Kyiv-Wilno, Kyiv-Kharkov or Kyiv-Budapest).

If the Kyiv station was already prescribed to a different route, you could still complete the ticket by activating the Kyiv-Bucaresti line through a station placed on Bucaresti instead.



As for the electronic version of the game, it does not immediately show tickets as complete if they rely on stations; since only one of the routes out of any given city can be used, and which route that may be is not decided until after the game is over (but before the score is calculated), the game doesn't know which route you want to use until the end of the game. It will give you credit for routes completed with properly-used stations when it tallies up your score at the end.