= Solution
The derivative convention uses <hypercharge> Y with $Q=T_3+Y$. For each <lepton> family $l=e,\mu,\tau$, the left-handed fields form an <SU(2)> doublet $L_l=(\nu_l,l)_L^T$, with $Y=-1/2$. Their <weak isospin> values are $T_3=+1/2,-1/2$ and their electric charges are $0,-1$. The right-handed charged <lepton> $l_R$ is an <SU(2)> singlet, with $T_3=0$, $Y=-1$ and charge $-1$. The minimal model used here has no right-handed <neutrino>. A sterile right-handed <neutrino>, if added as a singlet with $Y=0$, would have no <charged weak current> interaction. Antiparticles carry the conjugate representations and opposite electric charges.
Expanding $\bar L_l i\gamma^\mu D_\mu L_l$ supplies $-g\bar L_l\gamma^\mu A_\mu^a\sigma_aL_l/2$. The off-diagonal entries of $A^a\sigma_a$ are $\sqrt2W^+$ and $\sqrt2W^-$. With the <chiral projector> $P_L=(1-\gamma^5)/2$, one has $\bar\nu_L\gamma^\mu l_L=\bar\nu\gamma^\mu P_L l$. Thus the two <W boson> interactions are
$$
\boxed{\mathcal L_{W^+,l}=-\frac{g}{2\sqrt2}W_\mu^+\bar\nu_l\gamma^\mu(1-\gamma^5)l,\qquad\mathcal L_{W^-,l}=-\frac{g}{2\sqrt2}W_\mu^-\bar l\gamma^\mu(1-\gamma^5)\nu_l.}
$$
For <quarks>, separate rotations diagonalize the up-type and down-type mass terms. If their left-handed rotations are $U_{uL}$ and $U_{dL}$, the charged interaction contains the <CKM matrix> $V=U_{uL}^\dagger U_{dL}$:
$$
J_{\rm had}^\mu=\sum_{i,j}\bar u_i\gamma^\mu(1-\gamma^5)V_{ij}d_j.
$$
There is an implicit sum over <color charge>. This <quark mixing> permits transitions between different families, rather than just the paired fields in a single doublet.
Let $J^\mu=\sum_l\bar\nu_l\gamma^\mu(1-\gamma^5)l+J_{\rm had}^\mu$ and $\kappa=g/(2\sqrt2)$. At momenta much smaller than $m_W$, neglect derivatives in the heavy <W boson> equation. Its mass and source terms can be completed to a square:
$$
m_W^2W_\mu^+W^{-\mu}-\kappa(W_\mu^+J^\mu+W_\mu^-J^{\dagger\mu})=m_W^2\left(W_\mu^+-\frac\kappa{m_W^2}J_\mu^\dagger\right)\left(W^{-\mu}-\frac\kappa{m_W^2}J^\mu\right)-\frac{\kappa^2}{m_W^2}J_\mu^\dagger J^\mu.
$$
This explicitly performs <integrating out a charged vector boson>. The resulting <Fermi interaction> is
$$
\boxed{\mathcal L_{\rm eff}=-\frac{G_F}{\sqrt2}J_\mu^\dagger J^\mu,\qquad\frac{G_F}{\sqrt2}=\frac{g^2}{8m_W^2}.}
$$
The omitted terms contain derivatives and are relatively suppressed by momentum squared divided by $m_W^2$.
For the <leptonic pion decay>, <Lorentz covariance> restricts a vacuum-to-spin-zero current matrix element to a constant times $p^\alpha$. In the rest frame, rotations eliminate the spatial components. Under <parity>, the temporal vector current is even, whereas the vacuum is even and the <pion> is odd, so its temporal matrix element equals its negative and vanishes. The temporal axial current is odd and is allowed. This proves the <vacuum-to-pseudoscalar current selection rule>, rather than merely assuming an axial current. Define the <pion decay constant> by
$$
\langle0|\bar u\gamma^\alpha\gamma^5d|\pi^-(p)\rangle=if_\pi p^\alpha.
$$
Then the vector-minus-axial <charged weak current> has matrix element $-if_\pi V_{ud}p^\alpha$, up to an irrelevant overall phase convention. Consequently the <scattering amplitude> for the decay can be written
$$
\mathcal M=\frac{iG_Ff_\pi V_{ud}}{\sqrt2}\,p_\alpha\bar u_e(k)\gamma^\alpha(1-\gamma^5)v_\nu(q).
$$
Use $p=k+q$, the momentum-space <Dirac equation> $\bar u_e\not k=m_e\bar u_e$, and $\not qv_\nu=0$ for a massless <neutrino>. Since $\not q(1-\gamma^5)=(1+\gamma^5)\not q$, the neutrino contribution vanishes, leaving
$$
\mathcal M=\frac{iG_Ff_\pi V_{ud}m_e}{\sqrt2}\bar u_e(1-\gamma^5)v_\nu.
$$
The <fermion spin sum> is, using the supplied <gamma matrix trace identities>,
$$
\sum_{\rm spins}|\bar u_e(1-\gamma^5)v_\nu|^2=\operatorname{tr}\bigl[(\not k+m_e)(1-\gamma^5)\not q(1+\gamma^5)\bigr]=8k\cdot q=4(m_\pi^2-m_e^2).
$$
Here $(k+q)^2=m_\pi^2$, $k^2=m_e^2$, and $q^2=0$. The mass term in the trace vanishes; anticommuting $\gamma^5$ reduces the remaining trace to twice $\operatorname{tr}(\not k\not q)$. It follows that
$$
\boxed{\sum_{\rm spins}|\mathcal M|^2=2G_F^2f_\pi^2|V_{ud}|^2m_e^2(m_\pi^2-m_e^2).}
$$
Using <two-body decay phase space>, with final momentum magnitude $(m_\pi^2-m_e^2)/(2m_\pi)$, gives the <leptonic pseudoscalar decay width>
$$
\Gamma(\pi^-\to e^-\bar\nu_e)=\frac{G_F^2f_\pi^2|V_{ud}|^2m_\pi m_e^2}{8\pi}\left(1-\frac{m_e^2}{m_\pi^2}\right)^2.
$$
Thus the requested $m_e^2(m_\pi^2-m_e^2)$ factor is already present in the squared amplitude, and the phase space supplies one more threshold factor.
The physical explanation is <helicity suppression>. In the <pion> rest frame the two momenta are opposite. The massless antineutrino has positive <helicity>; zero total angular momentum would require the electron to have positive <helicity> too. The <charged weak current> instead creates a left-chiral electron, whose <helicity> is negative when its mass vanishes. A nonzero electron mass admits the required opposite <helicity> with an amplitude proportional to $m_e$. \b[The decay amplitude vanishes for $m_e=0$, and the rate is suppressed by $m_e^2$.]
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