Given, heat loss per second $Q_1$ through wire is proportional to surface area of wire.

$\Rightarrow \quad Q_1=k \pi r^2 L(\Delta T)$

where, $k=$ proportionality constant

and $\Delta T=$ temperature difference of wire and surroundings.

and heat generated per second is

$Q_2=I^2 R=\frac{I^2 \rho L}{\pi r^2}$

In steady state, $Q_1=Q_2$

$\Rightarrow k \pi r^2 L(\Delta T)=\frac{I^2 \rho L}{\pi r^2} \Rightarrow \Delta T=\frac{I^2 \rho}{k \pi^2 r^4}$

$\therefore$ Steady state temperature is independent of length of wire.