  Kemeny's constant of a simple connected graph $G$ is the expected length of a
random walk from $i$ to any given vertex $j \neq i$. We provide a simple method
for computing Kemeny's constant for 1-separable via effective resistance
methods from electrical network theory. Using this formula, we furnish a simple
proof that the path graph on $n$ vertices maximizes Kemeny's constant for the
class of undirected trees on $n$ vertices. Applying this method again, we
simplify existing expressions for the Kemeny's constant of barbell graphs and
demonstrate which barbell maximizes Kemeny's constant. This 1-separation
identity further allows us to create sufficient conditions for the existence of
Braess edges in 1-separable graphs. We generalize the notion of the Braess edge
to Braess sets, collections of non-edges in a graph such that their addition to
the base graph increases the Kemeny constant. We characterize Braess sets in
graphs with any number of twin pendant vertices, generalizing work of Kirkland
et.~al.~\cite{kirkland2016kemeny} and Ciardo \cite{ciardo2020braess}.
