I think writing things like $\lambda(nm), \lambda(\text{angstrom})$ is causing you to misunderstand your own notation. Let's do it properly from the beginning. We have:
\begin{align}
E &= \dfrac{hc}{\lambda}
\end{align}
Now, divide both sides by $1\text{eV}$, and plug in the values for $h,c$ and keep track of all units:
\begin{align}
\dfrac{E}{1\text{eV}} &= \dfrac{hc}{1\text{eV}}\cdot \dfrac{1}{\lambda}
= (1.24\times10^{-6} \text{m}) \cdot\dfrac{1}{\lambda} \tag{$*$}
\end{align}
Now, we can always write
\begin{align}
\lambda &= \lambda_{m} \cdot (1\text{ m}) = \lambda_{nm} \cdot (1\text{ nm}) = \lambda_{ang} \cdot (1 \, \text{ang}), \tag{$**$}
\end{align}
for some real numbers $\lambda_m, \lambda_{nm}, \lambda_{ang}$. Now, if we plug $(**)$ into $(*)$ then we find
\begin{align}
\dfrac{E}{1\text{eV}} &= \dfrac{1.24 \times 10^{-6} \text{m}}{\lambda_m \cdot (1\text{m})}
= \dfrac{1.24 \times 10^{-6} \text{m}}{\lambda_{nm} \cdot (1\text{nm})}
= \dfrac{1.24 \times 10^{-6} \text{m}}{\lambda_{ang} \cdot (1\text{ang})} \\\\
&= \dfrac{1.24 \times 10^{-6}}{\lambda_m}\,\,\,\, = \dfrac{1.24 \times 10^{3}}{\lambda_{nm}} \, \, \, \, \,\,\,\,= \dfrac{1.24 \times 10^{4}}{\lambda_{ang}}
\end{align}
Here is where I think you were making the big mistake: it is true that $1 \text{nm} = 10 \text{ ang}$, however, we have $\lambda_{nm} = 10^{-1} \lambda_{ang}$ (if your units get scaled in one way, then the coefficients have to get scaled in the opposite way in order to keep the product same). This is what I meant above when I said you misunderstood your own notation; you were probably thinking of your notation $\lambda(nm)$ as being a number $\lambda$ multiplied by $1\text{ nm}$, when it is really $\lambda_{nm} = \dfrac{\lambda}{1 \text{ nm}}$.
A remark about terminology:
The general rule is that if you have a physical quantity like energy $E$, and you say "energy in Joules", then what you mean is $\dfrac{E}{1\text{J}}$. If you say "energy in micro electron-volt" what you mean is $\dfrac{E}{1 \mu\text{eV}}$. So, for example, whenever you wrote $\lambda(angstrom)$ in your post, what you should have written is $\lambda = \frac{\lambda}{1 \text{ ang}} \cdot (1 \text{ ang})$, and this is exactly what I called $\lambda_{ang} \cdot (1 \text{ ang})$ in my answer above.
