Before we even start to talk about this, we should remember: this all happened on 2nd down with about a yard and a half to go!
It was close, but I see the ball was beyond the Neutral Zone when it was picked up by K after rolling between the LOS and the line to gain; so K could not advance it - and the play should have been blown dead when he possessed it. The K player simply advanced a dead ball.
SO, ASSUMMING TIME COULD STILL BE ON THE CLOCK, WHOSE BALL IS IT (OR SHOULD IT BE)? K's ball 4th down and a half yard to go; or R's ball at the dead ball spot?
Forgive me for snipping a lot of that text -- but assuming there was time remaining, R is awarded the ball at the spot of first touching, 1st and 10. The fact that K attempted the kick on 2nd down and not 4th has no bearing.
If K had recovered behind the LOS and not made the line to gain, it would be 3rd down for K.
For ultra fun fun goodie pot stirring time, did K still have at least 4 players numbered 50-79 on the line of scrimmage during that play? I actually think they do (#50, #52, #54 and #56 with long snapper #31 in under the exception). However, who just realized they should have thought of that because it was 2nd down?