Learn Hardcore Indonesian: Setelah saya kirim email, Ibu Dini membalas dengan kabar baik. - After I sent the email, Mrs. Dini replied with good news.

Both are correct. Saya kirim email is very common in everyday spoken Indonesian (it drops the meN- prefix), while saya mengirim email is the more formal, fully affixed form. In careful writing, prefer mengirim. Your sentence would be perfectly fine as: Setelah saya mengirim email, Ibu Dini membalas dengan kabar baik.

When should I use mengirim, mengirimkan, and mengirimi?

mengirim [thing] (kepada/ke [person]): the default, neutral form. Example: Saya mengirim email kepada Bu Dini.
mengirimkan [thing] (kepada/ke [person]): often interchangeable with mengirim; sometimes adds a slight “for someone” nuance or just sounds a bit more formal. Example: Saya mengirimkan email ke Bu Dini.
mengirimi [person] [thing]: focuses on the recipient; takes the recipient directly. Example: Saya mengirimi Bu Dini email.

Can I say Setelah mengirim email, Ibu Dini…?

Be careful: Setelah mengirim email, Ibu Dini membalas… normally implies that the one who sent the email was Ibu Dini, because the subject of the setelah-clause is understood to be the same as the main clause’s subject. To keep your original meaning (you sent it), say Setelah saya mengirim email or Setelah saya kirim email.

What does Ibu before a name mean, and why is it capitalized?

Ibu is an honorific for an adult woman (roughly Ms./Mrs./Madam). It shows respect and is capitalized when used as a title before a name: Ibu Dini. The short form Bu is also very common: Bu Dini. Lowercase ibu (no name) means “mother,” as in ibu saya.

Is Bu Dini okay, or should I stick to Ibu Dini?

Both are fine. Ibu Dini feels a bit more formal; Bu Dini is respectful but slightly more relaxed. Using just Dini is casual and usually for peers or close acquaintances.

Does Indonesian mark past tense? How do we know this happened in the past?

Indonesian doesn’t inflect verbs for tense. Time is inferred from context or time words. Setelah (“after”) already signals sequence, so past-ness is clear. You can add time expressions if helpful: Setelah saya mengirim email tadi pagi, …

Why kabar and not berita?

Kabar is “news/updates” in a personal sense (think Apa kabar?). Berita is “news” more in the media/public-announcement sense. For a personal reply with good news, kabar is the natural choice: kabar baik.

Why kabar baik, not kabar bagus?

Baik collocates with intangible things like behavior, intentions, and news. Bagus is for quality/appearance/performance of objects or results. The fixed, idiomatic phrase is kabar baik.

Is membalas the right verb for replying to emails? What about menjawab or merespons?

membalas (email/pesan/surat) = to reply to a message; very common for emails.
menjawab (pertanyaan/telepon) = to answer a question or a call.
merespons (sesuatu) or memberi respons = to respond (more formal/neutral, often used in business or official contexts).

Do I need to specify the object after membalas here?

Not necessarily. The object can be omitted if it’s obvious from context. If you want to be explicit, you can say Ibu Dini membalas email saya dengan kabar baik, or refer back with a pronoun: Ibu Dini membalasnya dengan kabar baik.

Can I move the setelah-clause to the end?

Yes: Ibu Dini membalas dengan kabar baik setelah saya mengirim email. When the setelah-clause comes first, use a comma after it; when it comes last, you normally don’t need a comma before setelah.

Could I use aku or the ku- prefix here?

Informal pronoun: Setelah aku kirim email, … or Setelah aku mengirim email, … are fine in casual contexts.
ku- attaches to the verb with no space: Setelah kukirim email, … This is correct but feels more literary/written than everyday speech.

Can I use sesudah instead of setelah?

Yes. Sesudah and setelah are near-synonyms. Both work here: Sesudah saya mengirim email, … Some people feel sesudah sounds a touch more formal or traditional, but they’re largely interchangeable.

Is email the best word, or should I use surel?

In everyday Indonesian, email is by far the most common. In official/government contexts you may see surel (from surat elektronik). Both are understood: mengirim email / mengirim surel.

Are there other natural ways to say “good news” here?

Yes. Kabar gembira is also common and a bit more emphatic/cheerful than kabar baik. Your sentence could be: … membalas dengan kabar gembira.