symmetry breaking - 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.

Answer

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

$$e = g \sin\theta_w = g' \cos\theta_w.$$