Kütüphaneye gelmişken bu kitabı da geri vereyim.

Because gelmek normally takes the dative case when it means to come to a place.

• kütüphane = library
• kütüphaneye = to the library

So:

• Kütüphaneye gelmek = to come to the library

The ending is -(y)e / -(y)a, the Turkish dative suffix. The extra y appears because the noun ends in a vowel.

Gelmişken means something like:

• since I’ve come
• now that I’m here
• while I’m at it, since I came

It often gives an opportunity or might as well feeling. The speaker is saying that coming to the library creates a good chance to do one more thing.

So the idea is not just I came, but more like:

• Since I’m already here, I may as well return this book too.

This is a very common question, because both forms can look similar at first.

gelirken

means:

• while coming
• on the way here

This focuses on the action of coming itself.

gelmişken

means:

• since I’ve come
• now that I’m here
• having already come

This focuses on the fact that the coming has already happened, and now the speaker wants to make use of that situation.

In this sentence, gelmişken is the natural choice because returning the book happens after arriving, not during the journey.

Not really, at least not in a strong way.

In many contexts, -miş can suggest:

• reported information
• indirect knowledge
• surprise

But in -mişken, that usual meaning is much weaker. The whole form acts more like a set expression meaning:

• since having done X
• now that X has happened

So in gelmişken, you should mainly understand the whole chunk as since I’ve come / now that I’m here, not as a special evidential meaning like apparently I came.

Because it is a specific direct object, so Turkish uses the accusative.

• bu kitap = this book
• bu kitabı = this book (as a specific object)

Here the speaker means a particular book: this book. Since it is definite/specific, Turkish marks it with the accusative.

That is why you get:

• bu kitabı geri vereyim = let me return this book

not a bare object form.

Here da means:

• too
• also
• as well

So bu kitabı da means:

• this book too
• this book as well

It is written separately because this da is a separate particle, not a case ending.

Compare:

• evde = in the house → -de is a suffix, so it is attached
• kitabı da = the book too → da is a separate word, so it is written separately

It appears as da rather than de because of vowel harmony.

Yes. Literally, geri vermek means to give back, and in everyday Turkish it is a very common way to say:

• to return a book
• to give back something borrowed

So for library books, geri vermek is completely natural.

A more formal alternative is iade etmek, but geri vermek is very common in normal speech.

Vereyim is a 1st person singular optative / volitional form. In natural English, it often means:

• let me return
• I’ll return
• I might as well return

It expresses the speaker’s own decision or suggestion to themself.

So:

• geri vereyim = let me return it / I’ll go ahead and return it

This is different from a plain future like:

• geri vereceğim = I will return it

Vereyim sounds more immediate and spontaneous, like the speaker is deciding on the spot.

The standard written form is vereyim.

It comes from:

• ver-
  • -(y)eyim

So the correct standard form is:

• vereyim = let me give / let me return

In casual speech, some people may pronounce it closer to veriyim, but vereyim is the standard spelling you should learn and use in writing.

Turkish often puts:

1. the context/background part first
2. the main verb at the end

So the sentence is structured like this:

• Kütüphaneye gelmişken = since I’ve come to the library / now that I’m here
• bu kitabı da = this book too
• geri vereyim = let me return

This is very natural Turkish word order. The sentence builds up the situation first, then ends with the main action.

You could move parts around for emphasis, but this version sounds smooth and idiomatic.

You could say something similar with gelince, but the nuance changes.

gelince

usually means:

• when I come
• once I come

It is more neutral and often points to a future or conditional situation.

gelmişken

means:

• since I’ve come
• now that I’m here
• while I’m at it

This has a stronger opportunity / might-as-well feeling.

So in your sentence, gelmişken is better because the speaker is taking advantage of already being at the library.