Branching fraction ratios between the decays $ {B}{(s)}^0\to J/\psi {\eta}^{\left(\prime \right)} $ are measured using proton-proton collision data collected by the LHCb experiment at centre-of-mass energies of 7, 8 and 13 TeV, corresponding to an integrated luminosity of 9 fb − 1 . The measured ratios of these branching fractions are $ {\displaystyle \begin{array}{c}\frac{\mathcal{B}\left({B}^0\to J/{\psi \eta}^{\prime}\right)}{\mathcal{B}\left({B}^0\to J/\psi \eta \right)}=0.48\pm 0.06\pm 0.02\pm 0.01,\ {}\frac{\mathcal{B}\left({B}s^0\to J/{\psi \eta}^{\prime}\right)}{\mathcal{B}\left({B}s^0\to J/\psi \eta \right)}=0.80\pm 0.02\pm 0.02\pm 0.01,\end{array}} $ B B 0 → J / ψη ′ B B 0 → J / ψη = 0.48 ± 0.06 ± 0.02 ± 0.01 , B B s 0 → J / ψη ′ B B s 0 → J / ψη = 0.80 ± 0.02 ± 0.02 ± 0.01 , where the uncertainties are statistical, systematic and related to the precision of the η (′) branching fractions, respectively. They are used to constrain the η/η ′ mixing angle, ϕ P , and to probe the presence of a possible glueball component in the η ′ meson, described by the gluonic mixing angle ϕ G . The obtained results are $ {\displaystyle \begin{array}{c}{\phi}{\textrm{P}}={\left({41.6}{-1.2}^{+1.0}\right)}^{\circ },\ {}{\phi}{\textrm{G}}={\left({28.1}_{-4.0}^{+3.9}\right)}^{\circ },\end{array}} $ ϕ P = 41.6 − 1.2 + 1.0 ∘ , ϕ G = 28.1 − 4.0 + 3.9 ∘ , where the uncertainties are statistically dominated. While the value of ϕ P is compatible with existing experimental determinations and theoretical calculations, the angle ϕ G differs from zero by more than four standard deviations, which points to a substantial glueball component in the η ′ meson and/or unexpectedly large contributions from gluon-mediated processes in these decays. The absolute branching fractions are also measured relative to that of the well-established $ {B}_s^0\to J/\psi \phi $ decay, which serves as the normalisation channel. These results supersede the previous LHCb measurements and are the most precise to date.