You can relate the change in angular momentum of the sphere to the change in linear momentum - they are related by the radius. That is one equation. The second equation tells you the relationship between angular velocity and linear velocity when the sphere stops sliding. Again there is a simple factor of $r$.
Finally you have the coefficient of friction so you know how large the force is. With the first two equations you get $F \cdot t$, and with the third one you find $t$ from $F$ .
Does that help?
Those two should be sufficient to solve this.
UPDATE
Here are the equations, and how to combine them:
Momentum equations:
$$ \begin{align}
\Delta p &= m \Delta v = F \Delta t\\
\Delta L &= I\Delta \omega = F r \Delta t\\
I &=\frac{2}{5}mr^2\\
\end{align}$$
Initial condition:
$$ \begin{align}
v &= v_0\\
\omega &= 0
\end{align}$$
Final condition (for rolling sphere):
$$ v = \omega r$$
Combining:
$$ \begin{align}
m\Delta v &= F \Delta t\\
m(v_0-v_f)&= F \Delta t &(1)\\
\\
I\Delta \omega &= F r \Delta t\\
I \omega _f&= F r \Delta t &(2)\\
\\
r \omega_f &= v_f
\end{align}$$
Eliminating $v_f$ from (1) using (2):
$$\begin{align}
m(v_0 - r \omega _f) &= F \Delta t \\
m(v_0-\frac{Fr^2}{I}\Delta t)&=F\Delta t\\
m v_0 &= F \Delta t(1 + \frac{mr^2}{I})\\
&=F \Delta t(1+\frac52)
\\
\Delta t &= \frac27\frac{mv_0}{m g u}\\
&= \frac27\frac{v_0}{gu}
\end{align}$$
Sanity check:
- If initial velocity is greater, skidding time increases
- If friction coefficient is greater, skidding time decreases
- If gravity is greater, skidding time decreases
All of these make sense.
