(4*L*w+24*u)/7L is not the correct answer paris.

The real answer is  (2*L*w+6*u)/5L .

One mistake you coulda done is that you havnt taken the I of the system as I of plank + I of ball.

Second mistake is that you changed the point about which yiu took angular momentum conversation initially and after collision. Angular momentum once fixed about a  point should not be altered even if it is conserved in all other points. Thats the only way someone will get your answer.

Peace

Yes you are absolutly right it requires a good understanding of conservation of momentum and Mechanical energy . The solution can be found out by applying conservation of angular monentum about the COM and Linear momentum ( why about COM ? ) the explanation is beyond scope of Highschool physics but all that can be said COM is very special Point and you can conserve angular monetum even when its moving (Like in case of rolling).

and two points

First the while apply conserbation of angular monetum about any Point "O" You have to find MOI about that point "O"

second very important Point in problem like this is .. if the object sticks to the rod, It changes the COM and the MOI as well