The memory of what had happened to her closest friend, and the only one she’d had for years, still haunted her to this day — and it felt like a bit of relief to share her feelings to his partner’s new partner.
Who is he? If you are still talking about Lapis, it should be "her partner’s new partner".

Anju, having had it enough as it after trying to stay out of it, spoke up at that point.
Do you mean "having had enough of it"? Continuing with "after trying to stay out of it" sounds still a bit clumsy. You may want to replace either "enough of it" by "enough of this" or "out of it" by something more specific like "out of the discussion" or "quarrel" or whatever "it" is.

"But t-that is … so horrible," Anju replied with a tearful expression, then turning to Aiko, determinedly said, "Aiko, don’t you think it is time to end this terrible feud of yours? You have alway been jealous of her, yet she has clearly suffered for more than its worth. "
alway should be "always".
The worth of what are you talking about? Did you mean "more than it is worth"? ("it is" may be contracted to "it’s", "its" is a possessive pronoun).
Also, the whole thing is a bit running on. I’d recommend to split the action between the spoken lines into two sentences.

"That would probably be a good idea," asked Binary.
How do you ask that?

And indeed, the one she had was Anju, her Digipartner.
This sounds weird. May there be a word missing?

Anju! It’s so good to see to see you again!
Too many words? I don’t believe that Aiko talks like that, even when she is excited.

It’s not that we are both on a break long enough to actually meet each other.
I’m not sure whether Anju means what I read here. "It’s not that ...", being a bit colloquial, implies that Anju currently is not on a break he deems long enough, and somehow (how?) also knows that Aiko is not on a long enough break, too, so that actually meeting each other is a really weird coincidence. I guess Anju means "It’s not usual that ..." or "It doesn’t happen often that ...".
And I guess that the period should actually be a comma, because the sentence is continued by "Anju replied ..." (that’s a really tricky punctuation rule because what was replied were actually several sentences).

And there is not much we can’t do since ...
Is the double negation intentional? There is actually a lot they can do? (Using double negations while meaning a single negation is significant for some dialects, but I don’t believe that Aiko talks like that.)

Still, we should enjoy this moment while it last. Where you headed?
Should be "lasts" (third person) and "Where are you headed?", although leaving "are" out may be an intentional colloquialism. (Again, I don’t believe that Aiko talks like this.)

I specially like seeing the ones we knew growing up and still there.
I’m starting to wonder whether Aiko has a habit of leaving "are" out. Maybe the sentence should better end after "growing up", because she obviously can’t see the ones who aren’t still there.
Also, MS Word Spell and Grammar Check thinks that specially is not the right choice of word, it should be "especially", but that’s beyond my understanding of this foreign language.

And even we do, it’s probably unlike that it would always on same day and time like this
Aiko continues to leave words out. This should probably be "And even if we do, it’s probably unlike that it would always happen on the same day and time like this". Also, unlike may need to be "unlikely".

We are both always get too many missions for that.
To compensate for Aiko, Anju inserts an "are" that doesn’t need to be there.

Yeah, me too. Hopefully we can find soon now.
And then, Anju leaves out an "it" (should be "find it soon").

Aiko hugged her right back, a warm smile showing itself on her face.
How does the smile do that? Actually, the face is doing the showing, so even "a warm smile showing on her face" is a bit weird, although not unusual ("her face showing a warm smile" would be more accurate); the word "itself" definitely should not be there.

I don’t particularly mind, if helps you with your class, but what is it that you would be talking about?
Another "it" missing (should be "if it helps you with your class").

She bit her lip nervously.
Who? Obviously not Lapis, the last female character mentioned. With so many female characters being around, figuring out who it is takes too long. Better just say "Diane".

Aiko, don’t think you played a part into that as well?
Should be "don’t you think".

Enjoying this was difficult, seeing how often I was forced to figure out what was said and done. Halfway through I was close to give up, but it got better from there.

HG

Okay, that's an interesting connectin to have between them, I guess. And having them called out by their Poke/Digi-partner was a nice touch too. (Will Anju have its own missions someday?)

And now, anybody can see where this next mission will go. How bad will the two of them screw up, though? anyways, spectacle is guaranteed.

I liked the interaction between the agents, and especially the way that Diane pointed a way towards some kind of reconciliation. Let's see if anything comes of it.

And more than that: I didn't spot any SPaG errors!

I look forward to seeing how Aiko and Lapis continue from here.


(OT: Speaking about agent interaction, paging SkarmorySilver:
I'm curious: What's Backslash's progress on teaching Cupid how to read?)