Learn Hardcore Japanese: kinou ha keeki wo hutatu tabete simaimasita. - Yesterday I ended up eating two pieces of cake.

(without は), it’s still correct and natural. The version with は slightly emphasizes “yesterday” as the frame of what you’re talking about, like you’re contrasting yesterday with other days or focusing on what happened specifically yesterday.

What exactly does ケーキを do in this sentence, and why is を used?

ケーキ is the direct object of the verb 食べてしまいました (ate up / ended up eating). The particle を marks the direct object—the thing that the action is performed on.

ケーキを食べる = to eat cake
本を読む = to read a book
水を飲む = to drink water

So ケーキを means cake (as the thing that was eaten).

Why is the counter 二つ used, and not something like 二個 or 二枚?

二つ uses the generic Japanese counter 〜つ, which is often used:

when you don’t know the specific counter,
when it’s not important to specify the exact type,
or in casual / general speech.

For cakes, there are more specific counters (depending on shape/serving):

二個（にこ）: two small roundish items (pieces, whole cakes, etc.)
二切れ（ふたきれ）: two slices
二枚（にまい）: two flat things (can be used for slices of cake or bread)

Using 二つ is like saying two (of them) without focusing on whether they’re whole cakes or slices. It’s common and perfectly natural.

What nuance does 〜てしまいました add compared to just 食べました?

〜てしまいました has two main nuances:

Completion / “all done”
- Emphasizes that the action is completely finished.
- 全部食べてしまいました。 = I ended up eating it all / I ate it up.
Regret / unfortunate result
- Often implies the speaker feels regret, surprise, or that it was not ideal.
- ケーキを二つ食べてしまいました。 can suggest
  I (kind of unfortunately) ate two pieces of cake.
  Maybe more than intended.

If you say ケーキを二つ食べました, it’s more neutral: I ate two pieces of cake, with no particular hint of regret or “oops” feeling.

Is しまいました a separate verb, or part of the verb 食べる?

Originally, しまう is an independent verb meaning to put away / to finish doing something. But in 〜てしまう, it acts like an auxiliary verb attached to the て-form of another verb.

Dictionary form: 食べてしまう
Polite present: 食べてしまいます
Polite past: 食べてしまいました

So 食べてしまいました is:
食べる (to eat) → 食べて (て-form) + しまいました (polite past of しまう)
Together, they form one unit meaning ended up eating / ate completely (with possible regret).

What grammatical form is 食べて here?

食べて is the て-form of the verb 食べる (to eat).

Dictionary form: 食べる
て-form: 食べて

The て-form is used to:

connect verbs: 食べて、寝る (eat and then sleep)
make requests: 食べてください (please eat)
attach auxiliary verbs: 食べてしまう, 食べている, 食べておく, etc.

In this sentence, 食べて is used to attach the auxiliary しまう → 食べてしまいました.

Can the word order be changed? For example, can you say ケーキを昨日二つ食べてしまいました?

Yes, Japanese word order is relatively flexible, especially for elements marked by particles. All of these are grammatically possible:

昨日はケーキを二つ食べてしまいました。
昨日ケーキを二つ食べてしまいました。
ケーキを昨日二つ食べてしまいました。
ケーキを二つ昨日食べてしまいました。

The most natural and neutral is probably one of the first two. Moving 昨日 or 二つ around can slightly change the rhythm or what feels emphasized, but the basic meaning stays the same.

You must, however, keep 食べてしまいました together at the end; you can’t split them up.

Could you say 昨日は二つケーキを食べてしまいました, with the number before ケーキ?

Yes, that’s also acceptable:

昨日は二つケーキを食べてしまいました。

Placing the counter before the noun is common in spoken Japanese, especially with 〜つ:

三つりんごを食べた。 (more natural as りんごを三つ食べた, but you do hear both)
二つケーキを食べた。

The most standard pattern in textbooks is:

ケーキを二つ食べてしまいました。

but 二つケーキを… is also used, especially in conversation.

What’s the difference between 昨日はケーキを二つ食べました and 昨日はケーキを二つ食べてしまいました?

Grammatically both are correct; the difference is nuance:

食べました
- Simple past, neutral.
- I ate two pieces of cake yesterday. (just stating the fact)
食べてしまいました
- Past + completion and/or regret.
- I ended up eating two pieces of cake yesterday.
- Suggests: I finished them all, or I kind of wish I hadn’t, or it was more than I meant to eat.

Context and tone decide which nuance (completion vs regret) is stronger.

Is 昨日は required, or can I just say ケーキを二つ食べてしまいました?

You can absolutely drop 昨日は:

ケーキを二つ食べてしまいました。
= I (regretfully) ate two pieces of cake.

Adding 昨日は just adds the time frame “yesterday”. Whether you say it or not depends on what information is already clear from context. Japanese often omits things that are understood.

Is 食べてしまいました polite, casual, or formal?

食べてしまいました is polite past.

Casual dictionary: 食べてしまう
Casual past: 食べてしまった
Polite present: 食べてしまいます
Polite past: 食べてしまいました

So the whole sentence is in polite style, suitable for talking to someone you’d normally use です／ます with (teachers, coworkers, acquaintances, etc.). For casual speech, you might say:

昨日ケーキを二つ食べちゃった。

(食べちゃった is the contracted form of 食べてしまった.)