How to solve equations for the MMSE receiver?
I doubt there is general closed-form solution. For the first equation, you essentially want to solve $$\mu=\sum_k \frac{1}{a_k+b_k \mu}$$ for $\mu$ which equals to search for the zeros of a $k+1$th order polynomial, which is impossible for $k&gt;3$. So, I fear you need to do a numeric solution. Once you have $\mu$ you can go for a simple calculation of $\kappa$. 

Regarding a numerical solution, here's a plot of $$\mu-\sum_k \frac{1}{a_k+b_k \mu}$$ which should become $0$ for the solution:

N = 7
d = np.random.rand(N)
K = N+1
M = 16
p = 1

f = lambda mu: mu - 1/(K-1)*sum(1/(M*p*d_i*(1-(K-1)/M+(K-1)/M*mu)+1) for d_i in d)

mu = np.linspace(-5,5,10000)
plt.grid(True)

plt.plot(mu, f(mu))
plt.ylim((-0.5,0.5))




So, there are multiple solutions. Maybe you need to pick to positive one?