This is a tough problem. I don't have an answer yet.
If you know the velocity $v_2$ of the CoM and the angular velocity $\omega_2$ which the 2nd cylinder ($C_2$) acquires from the collision, then it is relatively easy to work out the angular velocity when $C_2$ stops slipping and is purely rolling. The difficulty for me is finding $v_2$ and $\omega_2$. Specifically, how to deal with the friction from the ground during the collision.
Since the collision is elastic (coefficient of restitution $e=1$), the velocity of the CoM of $C_1$ will be $0$ after the collision, while that of $C_2$ will be the amount lost by $C_1$. The collision may also transfer some angular momentum. In the absence of external torques, this would also be conserved, so the angular velocity lost by $C_1$ would equal that gained by $C_2$. However, there is also friction from the ground acting on both cylinders during the collision. It is not clear to me what effect this will have.
The friction force from $C_1$ on $C_2$ acts downward. It tends to give $C_2$ some backspin, and it increases the normal reaction from the ground, thereby increasing this friction force on $C_2$. The friction force from $C_2$ on $C_1$ tends to reduce or reverse its forward angular velocity and decrease the normal reaction with the ground, reducing this friction force on $C_1$.
What is not clear to me is how to calculate the friction impulses on the cylinders from the ground. In Satvik Pandey's solution (see below) the weight of the cylinders is ignored, but I do not think this is justified.
It seems to me that increased friction with the ground will prevent $C_2$ from rotating during the collision, so that $C_2$ will emerge with no angular velocity.
The problem appears on the Brilliant website where the initial velocity of $C_1$ is given as $17.5m/s$ and its radius as $R=1m$. The correct answer is stated to be $a+b=503$. One solution has been posted by Satvik Pandey, but his answer is $a+b=20+3=23$. The person who posted the question (Karthik Kannan) has not posted a solution nor any comments. Possibly KK is a pseudonym for Deepanshu Gupta who apologises for getting the question wrong.
Satvik Pandey discussed the problem in Physics Forums where he provided the above answer.
In post #16 Satvik says this answer is wrong (presumably he learnt this from posting on the Brilliant site) but he is convinced by his friend's solution, which has the correct answer of $455+48=503$. The Homework Helpers both think this answer of $503$ is wrong and $23$ is correct. Satvik claims that in this solution the 1st cylinder rolls over the 2nd, but this is not consistent with coefficient of restitution $e=1$ ; the rolling contact point implies $e=0$.
I shall keep working on this problem and add to my answer when I make progress.