In order to determine the properties of single magnetic impurities in the Kondo state and the effects of these single magnetic impurities on the host-conduction-electron spin system, the Fe-impurity contributions to the ${\mathrm{Cu}}^{63}$-host nuclear-magnetic-resonance (NMR) linewidth $\ensuremath{\Delta}{H}_{i}$, and spin-lattice relaxation time ${T}_{\mathrm{li}}$, have been studied over a wide Fe-concentration range ($0<c<1260$ ppm) in $\mathrm{Cu}\mathrm{Fe}$. The NMR-linewidth measurements made from 1.65 to 77\ifmmode^\circ\else\textdegree\fi{}K and in magnetic fields from 2 to 16 kOe and in some cases up to 60 kOe, show the anomalous behavior of the slope $S=\frac{d\ensuremath{\Delta}{H}_{i}}{\mathrm{dH}}$ originally observed by Heeger et al. and studied for a 480-ppm $\mathrm{Cu}\mathrm{Fe}$ alloy by Golibersuch and Heeger exists over a wide Fe-concentration range. This anomalous behavior, which consists of the transition from a constant slope at low fields, ${S}_{L}$, to a smaller magnitude slope at high fields, ${S}_{H}$, occurs in a relatively narrow range of fields about some critical field ${H}_{c}$. This behavior clearly results from the single-impurity contribution to the NMR linewidth as evidenced by the linear concentration dependence of both ${S}_{L}$ and ${S}_{H}$ and also by the concentration independence of $\frac{{S}_{L}}{{S}_{H}}$. ${S}_{H}$ has the same ${(T+29)}^{\ensuremath{-}1}$ temperature dependence as the bulk susceptibility, while ${S}_{L}$ is enhanced for $H<{H}_{c}$ and $T<{T}_{c}\ensuremath{\approx}6$ \ifmmode^\circ\else\textdegree\fi{}K. At 1. 65 \ifmmode^\circ\else\textdegree\fi{}K, ${S}_{H}=(1.50\ifmmode\pm\else\textpm\fi{}0.10)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}c$, ${S}_{L}=(2.83\ifmmode\pm\else\textpm\fi{}0.10)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}c$ ($c$ in ppm), and $\frac{{S}_{L}}{{S}_{H}}=1.9$. These results show that the Ruderman-Kittel-Kasuya-Yosida-like oscillatory conduction-electron spin polarization existing about an impurity for $T\ensuremath{\gg}{T}_{K}$ is either enhanced for $T<{T}_{c}$ and $H<{H}_{c}$, or else an additional long-range oscillatory spin polarization is formed in the Kondo state. From the inverse concentration dependence of ${H}_{c}$ we conclude that long-range interactions of sufficient strength exist between Fe spins via the $d\ensuremath{-}d$ double-resonance mechanism to effectively saturate the extra oscillatory spin polarization in successively smaller applied fields as the Fe concentration increases. The impurity-induced host relaxation rate is linear in Fe concentration up to at least 300 ppm, decreasing from ${T}_{\mathrm{li}}^{\ensuremath{-}1}=2.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}c$ ($c$ in ppm) for 2.65 kOe to ${T}_{\mathrm{li}}^{\ensuremath{-}1}=2.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}c$ for 15 kOe at 1. 65 \ifmmode^\circ\else\textdegree\fi{}K. The low-concentration data follow a single curve when plotted as $T{(c{T}_{\mathrm{li}})}^{\ensuremath{-}1}$ vs $\frac{T}{H} (0.1\frac{^{\ensuremath{\circ}}\mathrm{K}}{\mathrm{kOe}}<\frac{T}{H}<1.0\frac{^{\ensuremath{\circ}}\mathrm{K}}{\mathrm{kOe}})$. Comparison of this curve with the existing high-temperature ($T\ensuremath{\gg}{T}_{K}$) theories would imply that the spin-lattice relaxation in the liquid-helium temperature range is dominated by a dipolar coupling of the nuclei to longitudinal dipolar fluctuations of the impurity spin. These results are discussed in the light of the ${T}_{\mathrm{li}}$ data for $T>{T}_{K}$ which does not appear to be consistent with this mechanism suggesting that none of the $T\ensuremath{\gg}{T}_{K}$ relaxation mechanisms may be simply extended to the region $T<{T}_{K}$.