Pattern Recognition (Bishop) - Maximum Likelihood
The partial derivative you have calculated is incorrect. You are probably confused by the index k in the equation.
Consider expanding the Langrangian:
$$ L = m_{1}ln\mu _{1}+ m_{2}ln\mu_{2} + ... + m_{K}ln\mu_{K} + \lambda(\mu_{1} + \mu_{2} + ... + \mu_{2} - 1) $$

Now, take the derivative with respect to $ \mu_{1} $

$$ \frac{\partial L}{\partial \mu _{1}} = \frac{m_{1}}{\mu _{1}} + \lambda $$

Then, in general,
$$ \frac{\partial L}{\partial \mu _{k}} = \frac{m_{k}}{\mu _{k}} + \lambda $$

Equate it to zero to get $ \mu _{k} = -m_{k} / \lambda $

Now substitute it in the constraint 
$$ \sum_{k=1}^{K}\mu_{k} = \sum_{k=1}^{K}\frac{-m_{k}}{\lambda} = 1 $$

Then $ \lambda = \sum_{k=1}^{K}-m_{k} = -N $

Finally, you get your answer
$$ \mu_{k} = \frac{m_{k}}{N} $$