Salep ve sütlaçtan sonra kimse kahve istemedi; sadece eniştem bir gofret daha aldı.

Sonra means after, but in Turkish it is a postposition, so it comes after the noun phrase, not before it.

The ending -tan is the ablative suffix, which often means from and is required before sonra.

So:

• sütlaç = rice pudding
• sütlaçtan sonra = after rice pudding

The form is -tan rather than -dan because sütlaç ends in the voiceless consonant ç, and Turkish matches the suffix to that sound.

In Turkish, when two nouns are joined with ve, the case ending is often placed only on the last item, and it applies to the whole phrase.

So Salep ve sütlaçtan sonra means after salep and sütlaç.

A more fully marked version would be:

• Salepten ve sütlaçtan sonra

But the original version is natural and common.

Because Turkish uses postpositions where English often uses prepositions.

English says:

• after salep and sütlaç

Turkish says:

• salep ve sütlaçtan sonra

So the idea is the same, but the word meaning after comes after the noun phrase.

This is normal Turkish grammar. In negative sentences, kimse is used together with a negative verb to give the meaning no one / nobody.

So:

• kimse ... istemedi = nobody wanted ...

This is similar to:

• Kimse gelmedi. = Nobody came.
• Kimse bilmiyor. = Nobody knows.

You can also hear hiç kimse for extra emphasis:

• Hiç kimse kahve istemedi.

Because kahve here is a non-specific direct object. In Turkish, non-specific direct objects usually do not take the accusative ending.

So:

• kahve istemedi = did not want coffee / any coffee

If you said kahveyi istemedi, it would usually mean a specific coffee:

• the coffee
• that coffee
• a coffee already known from the context

This is a very important Turkish distinction.

İstemedi breaks down like this:

• iste- = want
• -me- = negative
• -di = past tense

So:

• istedi = wanted
• istemedi = did not want

It is the simple past negative form.

Sadece means only or just.

In this sentence, it comes before eniştem, so it focuses on who did the action:

• sadece eniştem = only my brother-in-law

So the idea is:

• Nobody wanted coffee; only my brother-in-law took one more wafer.

If you move sadece, the focus can change:

• Eniştem sadece bir gofret daha aldı = My brother-in-law took only one more wafer.

The ending -m means my.

So:

• enişte = brother-in-law
• eniştem = my brother-in-law

Turkish often uses these possessive endings with family terms:

• annem = my mother
• babam = my father
• ablam = my older sister
• eniştem = my brother-in-law

Here daha means more in the sense of another / one more.

So:

• bir gofret daha = one more wafer / another wafer

This is a very common pattern:

• bir kahve daha = one more coffee
• iki dakika daha = two more minutes
• bir soru daha = one more question

So daha is not always the comparative more as in daha büyük = bigger; it can also mean another / additional.

For the same reason as kahve: it is a non-specific direct object.

• bir gofret aldı = he took/got a wafer, one wafer
• gofreti aldı = he took the wafer, a specific wafer

The word bir also helps show that the object is indefinite:

• bir gofret daha = one more wafer

Literally, yes: aldı is the past tense of almak, which basically means to take or to get.

But in food and drink contexts, it often sounds more natural in English as:

• got
• had
• sometimes ordered

So bir gofret daha aldı can be understood as:

• he got another wafer
• he had one more wafer

The exact English wording depends on context.

Because Turkish normally prefers subject–object–verb order, especially in neutral statements.

So:

• kimse kahve istemedi
• eniştem bir gofret daha aldı

both end with the verb.

Turkish word order is flexible, but putting the verb at the end is the most basic and expected pattern.

The semicolon links two closely related clauses:

• Salep ve sütlaçtan sonra kimse kahve istemedi
• sadece eniştem bir gofret daha aldı

It shows a stronger break than a comma, but the two ideas are still closely connected. In ordinary writing, you might also see these as two separate sentences.