Answer
$\rho(r)==6 K \varepsilon_0 r^3 $
Work Step by Step
We use
$
E(r)=\frac{q_{\mathrm{esc}}}{4 \pi \varepsilon_0 r^2}=\frac{1}{4 \pi \varepsilon_0 r^2} \int_0^r \rho(r) 4 \pi r^2 d r
$
to solve for $\rho(r)$ and obtain
$
\rho(r)=\frac{\varepsilon_0}{r^2} \frac{d}{d r} r^2 E(r)=\frac{\varepsilon_0}{r^2} \frac{d}{d r}\left[K r^6 \|=6 K \varepsilon_0 r^3 .\right.
$
