  We prove Torelli-type uniqueness theorems for both ALG$^*$ gravitational
instantons and ALG gravitational instantons which are of order $2$. That is,
the periods uniquely characterize these types of gravitational instantons up to
diffeomorphism. We define a period mapping $\mathscr{P}$, which we show is
surjective in the ALG cases, and open in the ALG$^*$ cases. We also construct
some new degenerations of hyperk\"ahler metrics on the K3 surface which exhibit
bubbling of ALG$^*$ gravitational instantons.
