Thank you @IgorKhavkine. I finally proved the required bounds for $P_{\ell}$ and $Q_{\ell}$ and completed the proof of the desired estimate. Thank you for all your help. :)

@IgorKhavkine I have tried what you recommended. The required estimates we need on $P_{\ell}$ and $Q_{\ell}$ seem to be difficult unless I'm missing something obvious. I edited the post to show my progress.

I have tried using Frobenius method and got all the coefficients, but the expression was very. But I didn't know that $(r^2-1)W(r)$ is a constant! This simplifies the expression significantly since there is no sum in the denominator! I will try this today. Thank you Willie and Igor.

Are there more robust techniques to prove this kind of estimate? I posted a similar question here with a similar ODE, but couldn't show it the same way.

Yes, I understand. I believe what you did above implies the estimate in the original post for any $\ell$. The hard part is to show that $C$ can be chosen independently of $\ell$ for large $\ell$. Thank you for helping me (you have also helped me several times in the past :) ).

Thank you! It's interesting that both the $C^0$ and $L^2$ norm of the forcing term appear.

$\ell$ is a positive integer. I want this estimate because it will eventually imply regularity results on solutions to the Poisson equation in $\mathbb{R}^3\setminus B_1$ equipped with an asymptotically flat metric $g$. In particular, if the forcing term has k angular derivatives in $L^2$ and $C^0$, then the solution and its derivative have $k+2$ and $k+1$ angular derivatives in $L^2$ and $C^0$. I haven't found such regularity results (where only angular derivatives are addressed) in the literature. (note the solution will have only 2 radial derivatives, but possibly much more angular ones.

@Hannes Thank you! That was what I was looking for. I guess "trace theorem" is not the right phrase to use.

Oh I didn't mean to add the "$u_1$". Thanks for pointing that out. I edited the post. I see, so we cannot make sense of $u(r)$ for a given $r$. If we in addition know that $\partial_r u \in L^2([0,1];H^{k-1})$, then is there a trace theorem saying something about $u(0)$?