Confusion with pronouns proceeding gerunds

So we took our time getting back, [him telling me how glad he was that he'd been able to give the woman what she deserved].

The verb "telling" is non-finite so it's a subordinate clause, here functioning as an adjunct.

Since it has a subject, "him", it is, more specifically, an 'absolute' construction, one that has no syntactic link to the main clause. Absolute clauses are supplements, which are not dependent on some other element in the sentence, so although the clause is subordinate, it is not a dependent one.

There is no internal marker of subordination: it is shown to be subordinate by virtue of its function in the larger construction.

With gerund-participial clauses in adjunct function, the choice of subject is between nominative "he" and accusative "him".

• So we took our time getting back, John telling me how glad he was that he'd been able to give the woman what she deserved.

Substituting a noun for the pronoun here shows that this is an absolute construction (here, the semantic relationship with the main clause being perhaps mere synchronicity, though arguably causal).

• So we took our time getting back, with John telling me how glad he was that he'd been able to give the woman what she deserved.

... is an alternative way of putting this. Here, 'John' has no case, though the preposition governs the accusative where available:

• So we took our time getting back, with him telling me how glad he was that he'd been able to give the woman what she deserved.

Arguing from analogy (often fatal!) we see that 'him' is the obvious choice in the posted original.