= Solution
The <London penetration depth> measures the density-to-mass ratio of carriers participating in superconducting screening. With electron-counted density $n_s$ and the stipulated bare <electron mass> $m_e$,
$$
\lambda_L^2=\frac{m_e}{\mu_0n_se^2},\qquad
n_s=\frac{m_e}{\mu_0e^2\lambda_L^2}.
$$
Using $m_e=9.109\times10^{-31}\,\mathrm{kg}$, $e=1.602\times10^{-19}\,\mathrm C$, $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{H\,m^{-1}}$ and $\lambda_L=22\times10^{-9}\,\mathrm m$ gives
$$
\boxed{n_s\simeq5.8\times10^{28}\,\mathrm{m^{-3}}.}
$$
This is the requested conduction-electron density within the assumed local <London equation> model. More precisely, it is the <superconducting carrier density from London screening>; identifying it with all conduction <Electrons> assumes that they participate in screening at the measurement <temperature>. No temperature-dependent normal-carrier fraction is supplied.
If the condensate is instead counted as <Cooper pairs>, the pair mass is $2m_e$ and charge magnitude is $2e$, giving pair density $n_{\rm pair}=n_s/2\simeq2.9\times10^{28}\,\mathrm{m^{-3}}$. Keeping <Electron> and pair counting consistent avoids a spurious factor of two.
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