Your working... assumes the answer.
Your first line in (3) is like "these two things are equal". That's not true. You want to show these things are equal.
Your working should go:
k.2^{k+1} + (k+1)2^{(k+1)-1} = \left (1 + (k-1)2^{k} \right ) + (k+1)2^{k} (by inductive assumption)
1...