  In this work we will use a general procedure to construct higher local
Hamiltonians for the affine $\mathfrak{sl}_2$ Gaudin model. We focus on the
first non-trivial example, the quartic Hamiltonians. We show by direct
calculation that the quartic Hamiltonians commute amongst themselves and with
the quadratic Hamiltonians which define the model. We go on to introduce a
certain next-to-leading-order semi-classical limit of the model. In this limit,
we are able to write down the full hierarchy of higher local Hamiltonians and
prove that they commute.
