jmc

Here we take advantage of the relationship between the sum and product of the roots of a polynomial and the coefficients of the polynomial.

If $\alpha ,\beta$ are the roots of the equation, then $k=\alpha +\beta$ and $\alpha \beta =-12$. Knowing that $\alpha \beta =-12$ and $\alpha ,\beta$ are integers, we can make a list of possible values for $\alpha$ and $\beta$. $$\begin{array}{r}(1,-12),(-1,12)\\ (2,-6),(-2,6)\\ (3,-4),(4,-3)\end{array}$$ The possible values for $k$ are $1-12=-11$, $12-1=11$, $2-6=-4$, $6-2=4$, $3-4=-1$, $4-3=1$.

Adding up the positive values of $k$, we get $11+4+1=\boxed{16}$.