Given $z(1+i)+i=0$, we first isolate $z$:

$z = \frac{-i}{1+i}$

To simplify this expression, we can multiply the numerator and the denominator by the conjugate of the denominator:

$z = \frac{-i(1-i)}{(1+i)(1-i)}$

Recall that the product of a complex number and its conjugate is the square of its magnitude, hence $(1+i)(1-i)=1^2+1^2=2$:

$z = \frac{-1-i}{2}$

Finally, separate real and imaginary parts to obtain:

$z = -\frac{1}{2} - \frac{1}{2}i$

Therefore, the correct answer is:
\boxed{B: -\frac{1}{2} - \frac{1}{2}i}