Learn English – Is it correct to say: “she’s having her housework done by this evening”

I would certainly change your sentence to "at that time".

What I believe you are missing is that "in" implies a duration and the phrase is actually a shortened version of "within" and skipping "amount of":

[with]in that [amount of] time I managed to undo the knot.

This means:

• I was given some amount of time
• I was able to complete the task

A longer version of the sentence, to put it in context, would be something like:

I was tied up in a closet and knew the bomb would go off in two minutes. In that time I managed to undo the knot, get out of the closet and escape the building just before it exploded.

And, if you were telling the story to someone, you might include "at that time":

A: Where were you yesterday evening?
B: I was tied up in a closet at that time and a bomb was going to go off in two minutes.

OK, here's the scoop on this usage: Please consider the utterance (I'm working with speech here).

A) the repetition of the preposition (stranding the preposition) is not needed:

1) declarative: I'm going to New York later this afternoon.

2) interrogative: Where are you going later this afternoon? Notice how the TO disappears in the question.

Ah, but then why do people say: Where are you going to [this afternoon]?

There are various answers. They are not educated speakers or they are sloppy speakers. The fact is that in the question to is not needed. That said, people do speak like that. Would a speaker like myself say it? Probably not. Except for emphasis: Where did you say you were going to?

Where are you going to?

Where to are you going?=not idiomatic or heard To where are you going?=not idiomatic

1) Declarative: He is at the game? [standard];

2) Interrogative: Where is he? The preposition, as with to above, is not needed.

3) Where's he at? or Where is he at? [marked as uneducated or dialectal, as in Black English or common varieties where people aren't really paying attention to their own speech]

At where is he? = not idiomatic, not heard

B) stranded prepositions (aka hanging prepositions)

Let's start with a declarative sentence: I'm writing about horror movies.

Interrogative (standard speech): What are you writing about?

Not repeating the object of the preposition is called preposition stranding and it is very common in spoken English and is used by native English speakers. /About what are you writing/ though grammatical would never be heard, really.

1) Declarative: This depends on the student's attitude:

2) Interrogative: standard speech: What does this depend on? More of a written form: On what does this depend? But, it could be said.

The last example: For whom are you doing this? Can be stranded in standard speech: Whom are you doing this for? Please note also that here /who/ is often used in place of /whom/ and is acceptable in some circles, but not in formal speech. However, the stranded preposition is fine.