Approximating an expression for a potential

There is nothing wrong with your first approximation. You got the leading term $2k/l$ correct. You just did not expand out far enough in powers of $x/l$ to see the $2k x^2/l^3$ term. If you were to keep going, you would find a term with $x^4/l^5$ and so on. You would need this if you wanted to study how the oscillation frequency varies with amplitude. How far you need to go depends on what effects you are after.

We have $$ x^2 \ll l^2 \implies \frac{2x^2}{l^2} \ll 2 \implies \frac{2x^2}{l^2} + 2 \approx 2 $$ So, $$ V \approx \frac{k}{l} \bigg(2 + \frac{2x^2}{l^2}\bigg) \approx 2\frac{k}{l} $$