Coupling constant in electroweak theory Electroweak theory has two coupling constants before and after Spontaneous Symmetry Breaking (SSB) each one for $SU(2)_L$ and $U(1)_Y$, though they are connected by Weinberg angle after SSB. My question is, how is unification complete with two independent couplings before SSB. The motive for unification is a single unified force with certain range and (coupling)strength. 
 A: The two parts of the electroweak gauge group do not separately describe the weak and the electromagnetic force.
The unified group here is not
$$ SU(2)_\mathrm{weak} \times U(1)_\mathrm{em}$$
but rather 
$$ SU(2)_L \times U(1)_Y $$
and the electric charge arises as a linear combination of hypercharge and weak isospin.
Therefore, although the group is not simple, the weak and electromagnetic forces have been unified, giving rise to two other forces. 
Edit: To adress the matter of coupling constants:
Indeed, before SSB there are two independent coupling constants $g'$ (for the $U(1)$) and $g$ (for $SU(2)$). One way to relate them to parameters after SSB is to think of the couplings constant $g$ vanishing, but a new parameter arising: the Weinberg angle. The Weinberg angle $\theta_W$ determines, what linear combination of of the neutral vector bosons $W_3$ from $SU(2)$ and $B$ from $U(1)$ turn into the massive $Z$ boson and what combination turns into the massless $\gamma$.
The Weinberg angle is determined through the gauge couplings as
$$ \cos\theta_w = \frac{g}{\sqrt{g^2 + g'^2}}.$$
In other words, in the breaking $SU(2)_L \times U(1)_Y$ both groups get broken, but there exists a linear combination of generators that remains unbroken. The $U(1)$ spanned by this generator does not relate 1:1 to either gauge group before SSB, though!
The coupling constant for the photon now relates to the couplings before SSB through the Weinberg angle
$$ e = g \sin\theta_w = g' \cos\theta_w.$$
A: Unification in the EW theory means that weak interactions that are short range are not very  difference at short distances than electromagnetism, both are described by spin-1 gauge boson exchange with couplings that are similar in size. They differ mainly  at large distances where the mass of the mediators becomes important, killing the interactions at distances $r>1/m_{W}$. In this sense, EW unification is just meant for the conceptual understanding of these two seemingly different forces that are instead described by very similar equations and coupling constants.
