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When Myrrhine learnt that her child was [= is] blind she said to her husband crying: Oh Zeus, what should we do? Pray to the gods to help us. And Dikaiopolis said: But we must carry the child to a doctor; but it's already getting evening. So now we must hurry to my brother's house and ask him to receive us. And tomorrow we must look for a doctor.

ἐπεὶ ἔμαθεν ὅτι....
when she learnt, came to know that... What simple, straightforward, specific fact?
- The fact that her son is deaf.
τί ἔμαθεν ἡ Μυρρίνη ἐν τῷ τετάρτῳ μαθήματι;
   ἔμαθεν ὅτι ἑορτὴν ποιήσουσιν οἱ  Ἀθηναῖοι.
ἐμάνθανες ξένας γλώττας ἐπεὶ παῖς ἦσθα;

   were you studying, did you use to study? (imperfect tense)
ἔμαθες ξένας γλώττας ἐπεὶ παῖς ἦσθα; 
Did you actually learn (simple fact) foreign languages? (aorist tense)
ὦ Ζεῦ, ἔφη. Oh God, she said.
For the conjugation (present/past) of  ἔφη see here.
τί ἔφη ἡ Μυρρίνη ἐπεὶ ἔμαθεν ὅτι λυκὸν ἀπέκτονεν ὁ παῖς;
   εὖ γε, ἔφη, μάλα ἀνδρεῖος εἶ. 
καὶ τί ἔφη ἡ Μέλιττα περί τοῦ Ὀδυσσέως;
   ὡς σοφός ἐστιν, ἔφη
τί ἔφη ὁ Δικαιόπολις ἐπεὶ μέγας λίθος ἐνέπεσε τῷ ποδί;
   ὦ Ζεῦ, ἔφη, φεῦ τοῦ ποδός.

τί δεῖ ἡμᾶς ποιεῖν; what are we do to? must/should we do?
τί δεῖ αὐτοὺς ποιεῖν;
δεῖ αὐτοὺς εὔχεσθαι τοῖς θεοῖς;
τίνα δεῖ αὐτοὺς ζητεῖν;

The impersonal verb δεῖ takes the accusative case (and not the dative as would be expected from the English translation: it is necessary for somebody to do something). Therefore this question could mean: Who must look for them? But here it quite obviously doesn't mean that. What does it mean and what is the answer?

διὰ τί δεῖ μένειν τὴν ὑστεραίαν;
παρὰ τίνα δεῖ αὐτοὺς σπεύδειν;
τί δεῖ αὐτοὺς αὐτὸν αἰτεῖν; 
ask, beg
δεῖ αὐτοὺς ταῦτα ποιεῖν 
      διότι νὺξ ἤδη γίγνεται 
night is falling or
      διότι νὺξ ἤδη γέγονε night has fallen
δεῖ ἡμᾶς αἰτεῖν αὐτὸν ἡμᾶς δέχεσθαι
We must ask him to take us in (for the night)
αἰτῶ τοὺς παῖδας ἡσύχους μένειν. 
   (I've got a headache andI ask them to be quiet, to remain still)
αἰτῶ σε μανθάνειν τὸν ἀόριστον (be busy learning, keep on at it)
αἰτῶ σε μαθεῖν τὸν ἀόριστον (learn it once and for all)
   AORIST infinitive for simple aspect: once and for all
ὁ Δικαιόπολις αἰτεῖ τὸν δοῦλον τοὺς λίθους λαμβάνειν 
asks him politely to take all those stone, keep taking them, progressive or repetitive aspect
κελεύει τὸν δοῦλον τὸν μέγαν λίθον λαβεῖν 
he orders him to take one specific stone, aorist for simple aspect
ὁ πάππος αἰτεῖ τὸν Φίλιππον τοὺς λύκους φεύγειν 
tells him to avoid wolves in general
ὁ πάππος αἰτεῖ τὸν Φίλιππον φυγεῖν τὸν λύκον μέγαν ὄντα 
tells him to run away NOW from this big wolf