If we cut a spring two halves (or consider a spring as two with the same lengths), each spring should have a doubled k.

Another view to hack the problem: Any part of a spring should be affected by the same force. Hence, the different force assumption should be fake.

Then, for detailed explanation: Think about the equation for the bottom end of the spring:$F=k\Delta l$. Then, we furtherly analyse the middle point of the spring, we have $F = k_{top\ half}\Delta x_{middle}=\frac12 k_{top\ half}\Delta l$. Since the "same force" theorem, $k_{top\ half}=2k$