When the two discs are joined together, their moments of inertia get added up.
Moment of inertia of the system of two discs, l = l1 + l2
Let ω be the angular speed of the system.
Total final angular momentum, Lf = (l1 + l2)ω
Using the law of conservation of angular momentum, we have:
Final kinetic energy Ef:
The loss of KE can be attributed to the frictional force that comes into play when the two discs come in contact with each other.