$1 UB MTT - BB, barely in the money, 2nd pair on the river

[Misfire]

I don't know much about MTTs, so this might be noobish. I checked a blah hand from the BB and was planning to check/fold after the flop, but once I caught my K on the river, I wasn't sure if should stab at it or not. I'm thinking not as only better hands would call me, but at the same time my stack is pretty desperate.

1000/2000 +200 ante

CO with 4125.
Button with 19030.
SB with 35030.
Hero with 12335.
UTG with 5415.
UTG+1 with 39160.
MP1 with 26370.
MP2 with 13510.
CO+1 with 44000.

SB posts the small blind of 1000.
Hero posts the big blind of 2000.

Hero: 7[img]/images/graemlins/spade.gif[/img] K[img]/images/graemlins/diamond.gif[/img]

Pre-flop:

UTG folds. UTG+1 folds. MP1 folds.
MP2 folds. CO+1 folds. CO
folds. Button calls. SB folds. Hero
checks.

Flop (board: T[img]/images/graemlins/club.gif[/img] 4[img]/images/graemlins/diamond.gif[/img] A[img]/images/graemlins/heart.gif[/img]):

Hero checks. Button checks.

Turn (board: T[img]/images/graemlins/club.gif[/img] 4[img]/images/graemlins/diamond.gif[/img] A[img]/images/graemlins/heart.gif[/img] 9[img]/images/graemlins/spade.gif[/img]):

Hero checks. Button checks.

River (board: T[img]/images/graemlins/club.gif[/img] 4[img]/images/graemlins/diamond.gif[/img] A[img]/images/graemlins/heart.gif[/img] 9[img]/images/graemlins/spade.gif[/img] K[img]/images/graemlins/club.gif[/img]):

Hero checks. Button checks.