  Given a genus $2$ curve $C$ with a rational Weierstrass point defined over a
number field, we construct a family of genus $5$ curves that realize descent by
maximal unramified abelian two-covers of $C$, and describe explicit models of
the isogeny classes of their Jacobians as restrictions of scalars of elliptic
curves. All the constructions of this paper are accompanied by explicit
formulas and implemented in Magma and/or SageMath. We apply these algorithms in
combination with elliptic Chabauty to a dataset of 7692 genus $2$ quintic
curves over $\mathbb{Q}$ of Mordell-Weil rank $2$ or $3$ whose sets of rational
points have not previously been provably computed. We analyze how often this
method succeeds in computing the set of rational points and what obstacles lead
it to fail in some cases.
