A square with area $4$ is inscribed in a square with area $5$, with each vertex of the smaller square on a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. The original answer is in the form \frac{k}{m}, where \frac{k}{m} is a fully reduced fraction. Please find the value of k + m.

Let's think step by step and output the final answer within \boxed{}.
user