I was looking out of the library's window when I saw my friend Sofía.

The possessive ('s) becomes redundant here, so there is no need to use it, you can just say "library window". For example;
"I looked out of the door's peephole." Can just be -> "I looked out of the door peephole."
"I opened the car's door." -> "I opened the car door."
As a general rule, when using the possessive, I would try to first read the sentence without it ('s) then with it, and see if this extra detail is necessary or redundant to avoid this error!

I felt very sleepy before I selected whatpicked (out) what I wanted to eat, so I only ordered a coffee and a small cake.

Picked out -> colloquial/casual. "She picked out/chose the pink top to wear to the date." I think that 'selected' is quite formal here, and because your other prior sentences are not too formal in style, to keep it consistent, I would stick with colloquial/less formal verbs!

Adding "only" to "so I ordered" adds a layer of reasoning to why you mentioned that you were sleepy, and makes sense for why you said 'so'. 'So' is like -> así que -> so like therefore. You were experiencing x, therefore you did y.

Adding 'only' has a specific purpose, and gives reasoning. "Because I was tired, I only ordered a coffee and small cake." It is very common to see in narrative writing and stories, used in this same way! It implies that you would have ordered more, but you were too tired (which is what I assumed your reasoning was). If the reasoning, however, was that you only ordered what you did because you were too sleepy to have a thorough look through the menu, you could have said: "I felt very sleepy before I picked out what I wanted to eat, so I only ordered a basic/simple coffee and a small cake." Adding 'basic' here then implies the other potential reasoning that you didn't have a thorough look through the menu.

Other examples:

"I only got 3 hours of sleep last night, so I only did 1 hour of study." (Implying that they would have done more study if had gotten more sleep)
"I only knew her through another friend, so we only spoke about basic things." (Implying that if you had known the person more, you would have spoken about more detailed things/more personal things)

It was late whenby the time (that) we stopped talking and eating, so we said goodbye to each other and then decided to go home.

"by the time" is a very colloquial phrase to use instead of 'when'. Makes your writing very natural.

Other examples:
"By the time I got home, the sun had already set." "It was already 5PM by the time we got home."

She felt delighted until we needehad to go.

"Had" is slightly more casual and is definitely more used than "needed to go". Because "needed to..." implies a bit of stress/emphasises obligation than "had to".

"He needed to study until 5PM, otherwise he would have failed the test."
"We had to eat at an outside table at the restaurant because there were no tables left inside."
^ the differences are the level of necessity with the usage!

We'd have liked to talk for more time!

I like this sentence, the use of the conjunction for 'we would' is very casual and natural. Well done!

Suddenly, she came back fastrushed back to the restaurant, looking a bit ashamed and she told me "I forgot to pay.".

- "she came back a bit fast" can just be "she rushed back to/to/from the restaurant" (more natural)
- "looking a bit ashamed" the 'looking' adds a bit of cohesion but still maintains the casual narrative style!
- Just punctuation; the (.) full stop always goes outside of the speech marks, not within!

"Don't worry, I had already done it by the time you headed off the coffee shop/took off."

- No need to say the place (coffee shop) because this is implied; the readers know that the only place you and your friend have been to, and would have had to paid money, is at the coffee shop!
- Also, I see that you used 'by the time' here, it is used very well, well done!
- "heading off" is generally used for somebody going to a place, instead of leaving from a place. "Are you heading off to the cinema now?" But I have seen it used for leaving a place, in just the form of, "Are you heading off?" < just as this sentence is, in speech with friends! You can use both though, the reader will understand, I think some uses are just a tiny bit more correct than others :)

But I'll be the one who pays the next time."

I know in Spanish that you would generally use the article for 'the next time', however in English, when talking to friends casually, we don't use 'the' unless for emphasis:

Person 1: "Sorry I missed the meeting again." Person 2: "This is the second time you missed it! Are you sure that you're going to be there the next time?"

Person 1: "I don't usually go to these types of things. I really loved meeting you at this Spanish meetup." Person 2: "Me too! And I know that it is your first time here, but, I always go, so I hope I see you next time!"

I had been waiting for thise moment to tell her how grateful I am to be her friend and that I missed her a lot.

"This moment" implies a planned moment, e.g. you planned to meet a friend at 5PM. You then would say to that friend when meeting them "I had been waiting for this (moment) all day!"

"The moment" implies a general moment, it can be any time, unplanned, but it is something that is likely to happen.

"I was waiting for the moment to tell you that I loved you." "I was waiting for the right moment to tell you that I am sorry." << your wishes are the same in both situations, however in one situation the moment is expected/agreed upon/anticipated/exact, whereas 'the moment' is unplanned, but likely to happen given if you remember in that moment, or there is an opportunity! I hope this makes sense!

In your example, you hadn't planned in advance "As soon as I meet up with my friend, I am going to tell her x." or, during your conversation, you didn't think "As soon as we finish eating, I am going to tell her how much I x." <<< these would then need "this moment" when the situation came up, because you were waiting for a specific moment to happen/the moment was premeditated.

Until that moment, I didn't realised that she felt the same way.

So basically, "until that moment" is the time before the moment. So, before talking to her, you didn't think she felt the same way. Then, after the moment, now you realised that she does. So, when you say, "Until that moment, I didn't realise that I didn't realise that X", X = the new fact/discovery.

"Before I met you, I didn't realise how unlucky I was."
"Before this job, I didn't realise how slow I was."

When she told me that she was happy to meet with me as well, I couldn't help but smile.

"To meet with me" -> physically meet up with somebody, any relationship
"to meet me" -> to meet somebody for the first time

"I couldn't help but smile." -> well done, this is so natural!

Finally, wWhen we had finished talking, we took a picture of uourselves in order to remember that special day.

No need to say finally here; "The waiter finally gave us our food." "A table finally became free." << implies waiting, impatience to an extent.

"We took a picture of ourselves" "We spoiled ourselves"

I was looking out of the library's window when I saw my friend Sofía.

This goes for any time you are referring to parts of a building.

I talked towith her and we decided to go to a cafe.

"to" makes the discussion sound one sided when it probably wasn't.

I felt very sleepy before I selectedchose what to eat, so I ordered a coffee and a small cake.

"Chose" is more natural sounding than "selected".

It was late when we stoppfinished talking and eating, so we said goodbye to each other and then decided to go home.

"Finished" is more natural sounding than "stopped". For "then decided to go home" in this situation as you and Sofía are going separate ways, it would be contextually slightly more appropriate to reference this.

Suddenly, she came back fastquickly looking a bit ashamed and she told me: "I forgot to pay.".

"looking" adds clarity and you can also redact the second mention of Sofía's pronoun as you're still on the same sentence. The addition of a colon and rearranging of quotation mark and period are corrections for this sentence's formatting.

"Don't worry,. I had already done itpaid the bill by the time you headed off from the coffee shop."

Don't worry stands as its own sentence and thus a period is more appropriate. "already done it" is easy to understand through inference, however, it is best to clarify the action you took (paying the bill).

But I'll be the one who pays thefor next time."

Finally, when we had finished talking, we took a picture of us together in order to remember that special day.

I was looking out of the library'sin the library looking out the window when I saw my friend Sofía.

Your sentence is correct, but the phrase "library's window" sounds strange, but I'm not sure why...The corrected version sounds more natural in my opinion.

I felt very sleepy before I selectedchose what to eat, so I ordered a coffee and a small cake.

This is also correct, but "chose" sounds more natural here.

She felt delighted until we needed to go.was a little sad to have to leave;

Wwe'd have liked to talk for more time!

Suddenly, she quickly came back fast, a bit ashamembarrassed, and she told me, "I forgot to pay."

"Ashamed" is a bit strong for this situation; better is "embarrassed." There are always commas before direct quotations.

"Don't worry, I had already donetaken care of it by the time you head offleft the coffee shop."," I said.

Just some vocabulary substitutions to make it sound more natural.

I had been waiting for this moment to tell her how grateful I amwas to be her friend and that I missed her a lot.

UntilAt that moment, I realised she felt the same way.

Finally, when we (had) finished talking, we took a picture of uourselves in order to remember that special day.

The pluperfect is correct here, but the simple past is also correct and sounds a bit more natural.