A couple of things: 1) It's a fallacy to think that kings are captured to end the game - it's sometimes common among very beginners who've not learned the game properly. Understandable, but: 2) A checkmate is if the king is under attack and there is no legal move to get out of check. The game ends there and the king is not taken.
So, in your situation, I"m choosing to ignore the actual location of the pieces - that's not your point. The fact that there is no legal move at all is beside the point. If you're in check and you can't move out of check by moving, blocking, or taking a piece (or any other legal move), you've lost the game by checkmate.
Getting more technical, It's also not a question of whose move it is. White just moved, period. It can't be White's move now since the previous move was a checkmate which wasn't identified and whatever move black made was an illegal move by not moving out of check.
Actually, in theory, a postion of checkmate which came from an illegal move or an illegal move several moves ago would nullify the checkmate. If it's recent, they'd put the pieces back to where they were and continue.
Situation below (I made it up quickly, I know its not feasible as is, but I'm sure real position could be thought up with time) What is the result?
White has the king pinned, check-mate, but it's black's move and there are none available. Isn't it then a draw? The real question is, does the game end at checkmate, or one more later with a king capture? Curious what people think. Thanks!