= Solution
When $kd_i\gg1$, expand the square root in the exact frequencies. The fast branch has
$$
\Omega_{\rm fast}=|q|\left(kd_i+\frac1{kd_i}+O((kd_i)^{-3})\right),
$$
while the slow branch has $\Omega_{\rm slow}=|q|[(kd_i)^{-1}+O((kd_i)^{-3})]$. Hence
$$
\boxed{\omega_{\rm fast}^2\simeq k_\parallel^2v_A^2k^2d_i^2,\qquad
\omega_{\rm slow}^2\simeq\frac{k_\parallel^2v_A^2}{k^2d_i^2}.}
$$
The fast dispersive wave is the mode sought in the question's subsequent fast-time limit. In a strongly oblique wave, $k\simeq k_\perp$, so its frequency scales as $|\omega|\simeq|k_\parallel|v_Ak_\perp d_i$. The slow branch instead tends to $(|k_\parallel|/k)\Omega_i$, where the <ion> <cyclotron frequency> is $\Omega_i=v_A/d_i=eB_0/(m_ic)$.
There is a terminology qualification: in the pressure-free incompressible Hall equations used here, the fast branch is conventionally a <whistler wave>. The question calls the requested small-scale waves <kinetic Alfvén waves>, but a general kinetic-Alfvén dispersion includes <pressure>/kinetic ordering absent from these equations. The boxed dispersion is the result for the printed model; no extra <pressure> response or kinetic term has been silently added.
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