$C^*$-algebras constructed from circle-valued two-cocycles on discrete abelian groups give many examples that have appeared frequently in noncommutative geometry, noncommutative tori being just one such example. In this talk, two ways are discussed to construct spectral triples on such twisted group $C^*$-algebras that are inspired by using length functions on the groups, as first was done by A. Connes in 1989. We also use characterizations of length functions given by M. Christ and M. Rieffel in 2017, where seminorms derived from certain length functions allow us to consider the twisted group $C^*$-algebras as quantum compact metric spaces. Spectral triples constructed as inductive limits of spectral triples on inductive sequences of $C^*$-algebras, as first considered by R. Floricel and A. Ghorbanpour in 2019, are of particular interest to us. New necessary and sufficient conditions for the convergence of an inductive sequence of metric spectral triples for the Gromov-Hausdorff propinquity in the sense of F. Latremoliere are given, and applied to construct metric spectral triples associated to length functions on discrete groups where the $C^*$-algebras correspond to noncommutative solenoids and certain $C^*$-algebras of Bunce-Deddens type. This talk is based on joint with C. Farsi, T. Landry, N. Larsen and F. Latremoliere.