You have a linear ODE with explicit coefficients: the solutions can actually be written down explicitly via [variation of constants](https://en.wikipedia.org/wiki/Variation_of_parameters). To summarize the result, let $g_{a,b}(r)$ be given by

$$ g_{a,b}(r) = \frac{a}{r^{\ell+1}} + \frac{b}{r^2} $$

you find that 

$$ r^2 g_{a,b}'' + 2r g_{a,b}' - \ell(\ell+1) g_{a,b} = -\frac{(\ell+2)(\ell-1) b}{r^2} $$

As 

$$ g_{a,b}(1) = a + b , \qquad g'_{a,b}(1) = -(\ell+1) a - 2b $$

for $g_{a,b}$ to solve the equation you indicated requires $(\ell+2)\ell-1)b = g_{a,b}(1) + g'_{a,b}(1)$. So we need (when $\ell > 0$)

$$   - \ell a - b = (\ell+2)(\ell - 1)b \implies a = \frac{1 - \ell - \ell^2}{\ell} b $$

which yields

$$ \alpha = \frac{1-\ell^2}{\ell} b, \qquad \beta = - \frac{b}{\ell}(1 + 2\ell - 2\ell^2 - \ell^3) $$

so the Dirichlet to Neumann map has norm exactly 

$$ \frac{\ell^3 - 2\ell^2 + 2\ell + 1}{\ell^2 - 1} = O(\ell) = O(\sqrt{\ell(\ell+1)})$$