Sometimes it is fun(!?) and useful to practice disecting a verse and making sure you know what all the individual parts are doing, by writing down everything you possibly can. He’s my attempt for Luke 1:57:
Τῇ δὲ Ἐλισάβετ ἐπλήσθη ὁ χρόνος τοῦ τεκεῖν αὐτὴν καὶ ἐγέννησεν υἱόν
Δέ – initial postpositive conjunction. Functioning to make a break from what has preceded. V57 starts a new section which tells the narrative of the birth and naming of John the Baptist (JtB). The previous section was the coming together of Elizabeth, Mary, JtB and Jesus and marked the end of the ‘accouncement’ narrative. Here we have the start of the actual birth narratives.
ὁ χρόνος – the nominative masculine singular subject of the verb ἐπλήσθη.
τοῦ τεκεῖν αὐτὴν – an articular infinitive – genitive masculine/neuter singular definite article with aorist active infinitive of τίκτω. I am not sure how this infinitive is functioning. Given it’s location, it seems like a adjectival genitive epexegetical infinitive (“the time of…“). But this is an unusual use of the infinitive, and one would expect a participle here (e.g. “the time of her giving birth“.). Alternatively, the infinitive could be adverbial articular infinitive (probably of result, possibly purpose). This is how Leedy takes it in the BW8 diagrams. Hence “The time was fulfilled for [with the result that] her to give birth…”. The αὐτὴν is the accusative subject of the infinitive.
Τῇ…Ἐλισάβετ – dative feminine singular article with an indeclinable proper noun, the phrase acting as the indirect object of the main verb (ἐπλήσθη)
ἐπλήσθη – 3s aorist passive indicative from πίμπλημι.
Putting it all together, we get a verb clumbsy translation like: “Now the time was fulfilled to Elizabeth for her to give birth…”
There is a second clause at the end of the sentence:
καὶ – conjunction linking this clause closely with what came before and moving the narrative forward.
ἐγέννησεν – 3s aorist active indicative from γεννάω. The subject is implicit and clarified by the content: Elizabeth.
υἱόν – masculine singular accusative direct object of preceding verse.