  We report the observation of the $B$ meson decay $B^\pm\to J/\psi \eta K^\pm$
and evidence for the decay $B^0\to J/\psi \eta K^0_S$, using {90} million
$BBbar$ events collected at the $\ensuremath{\Upsilon{(4S)}}\xspace$ resonance
with the $BaBar$ detector at the PEP-II $e^+ e^-$ asymmetric-energy storage
ring. We obtain branching fractions of $\cal{B}$$(B^\pm\to J/\psi \eta
K^{\pm}$)=$(10.8\pm 2.3(\rm{stat.})\pm 2.4(\rm{syst.}))\times 10^{-5}$ and
$\cal{B}$$(B^0\to J/\psi\eta K_{\rm{S}}^{0}$)=$(8.4\pm 2.6(\rm{stat.})\pm
2.7(\rm{syst.}))\times 10^{-5}$. We search for the new narrow mass state, the
X(3872), recently reported by the Belle Collaboration, in the decay $B^\pm\to
X(3872)K^\pm, X(3872)\to \jpsi \eta$ and determine an upper limit of
$\cal{B}$$(B^\pm \to X(3872) K^\pm \to \jpsi \eta K^\pm$) $<7.7\times 10^{-6}$
at 90% C.L.
