Answer
The spring constant of each half-spring is twice that of the original spring.
The frequency is increased by a factor of $\sqrt2$.
Work Step by Step
A spring of length L can be considered to be two springs in a series, each of length L/2. When a given force is applied to the whole spring, each half-spring feels the same force, but stretches only half the length that the entire spring stretches.
By the definition of spring constant, each half-spring has twice the spring constant of the original spring.
The frequency f is $\frac{1}{2 \pi}\sqrt{\frac{k}{m}}$. The spring constant is doubled, so the frequency is increased by a factor of $\sqrt2$.
