There is a missing black pawn on the side of the table. This was done to make the illusion that black was winning but in reality the material is tied.
In the closing credits, Tyler Hynes' character is erroneously shown as Joshua Wheeler (the character's last name is not mentioned in this episode). In all other appearances, he is known as Joshua Donovan, older brother of series regular Claudia Donovan.
When Leena is sitting by the chess table there is a clear view of the board positions. The combination shown would not require "3 months" to analyse. White can keep attacking, starting with Claudia's sacrificial (and obvious) Queen check. Checkmate can be forced in less than "nine moves".
But there is a discrepancy in the total of Black pawns on and off the board. Possibly one is off board and hidden by the Rook.
Also while Leena and Artie are talking the White Queen repeatedly vanishes and returns. When Claudia comes out the White Queen isn't there, but she then moves it to call check.