The initial angular momentum of two discs is I1ω1+I2ω2=I(ω1+ω2).
The final angular momentum of system is (I1+I2)ω=2Iω
From the conservation of angular momentum, ω=2ω1+ω2.
Initial kinetic energy, KEi=21I(ω12+ω22)
and final kinetic energy, KEf=21(2I)4(ω1+ω2)2
Loss in kinetic energy is ΔKE=21I1+I2I1I2(ω1−ω2)2
=21(2I)I2(ω1−ω2)2
=41I(ω1−ω2)2