= Solution
Use the standard Earth radius $a_E=6.371\times10^6\,\mathrm m$ and rotation rate $\Omega=7.292\times10^{-5}\,\mathrm{s^{-1}}$. At $10^\circ$S,
$$
f_0=-2.53\times10^{-5}\,\mathrm{s^{-1}},\qquad
\beta=\frac{2\Omega\cos10^\circ}{a_E}=2.25\times10^{-11}\,\mathrm{m^{-1}\,s^{-1}}.
$$
Take a predominantly zonal wave train, $l\simeq0$, and the gravest baroclinic mode from part (c). Then
$$
L_d=\frac{HN}{\pi|f_0|}=1.59\times10^5\,\mathrm m,\qquad
k=\frac{2\pi}{1250\,\mathrm{km}}=5.03\times10^{-6}\,\mathrm{m^{-1}},\qquad kL_d\simeq0.799.
$$
The background current is westward, so $U=-0.025\,\mathrm{m\,s^{-1}}$. The pulse uses the <baroclinic Rossby-wave group velocity>, not the crest speed:
$$
c_{gx}=U+\beta\frac{k^2-L_d^{-2}}{(k^2+L_d^{-2})^2}\simeq-0.102\,\mathrm{m\,s^{-1}}.
$$
Therefore
$$
\boxed{t_{\rm crossing}\simeq\frac{10^7\,\mathrm m}{0.102\,\mathrm{m\,s^{-1}}}\simeq9.83\times10^7\,\mathrm s\simeq3.1\,\mathrm{yr}.}
$$
For comparison $c_{px}\simeq-0.373\,\mathrm{m\,s^{-1}}$ would give about $0.85$ years, but that is a crest transit time and not the requested pulse delay.
The approximately three-year group delay is comparable to the stated three-to-five-year <El Niño–Southern Oscillation> period. Wind-driven thermocline disturbances can store and convey oceanic adjustment westward; interaction with the western boundary and subsequent faster signals can contribute a delayed feedback on the tropical coupled system. This calculation supports a relevant time scale, not a complete ENSO oscillator: realistic meridional structure, variable stratification and currents, damping, boundary conversion and equatorial coupling are omitted. A nonzero unspecified meridional wavenumber would also change the numerical group speed, so the $l\simeq0$ assumption is material.
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