Well, before anything else, let's address that by killing is not a form of the verb kill. It's a prepositional phrase that employs a form of the verb kill. It is wrong for this question the same way
He committed a crime and, while driving to purchase a frozen burrito at the Circle K three days afterwards, he felt a sudden bump as though he had hit a pothole or a small animal which he might have killed, possibly a bird.
is wrong. It's grammatically correct but completely outside the scope of acceptable answers based on the wording of the question.
There are lots of prepositions, conjunctions, &c. that work just fine here. By killing, in killing, through killing, of killing, without killing, beyond killing all work just fine. So do like killing and such as killing. Some of them mean the death is the crime; some of them mean the death is separate from the crime; some mean the death never happened. It's not important. They're all equally irrelevant to the question being asked.
He committed a crime _____ (kill) a bird.
wants some form of the verb kill. Your choices are kill, kills, to kill, killed, killing, and (probably) any of an utter horde of English modal constructions like will kill, about to kill, be about to kill, was about to kill, might have been about to kill, &c. with the exception of Southern English constructions like might've ought've could've been a-fixing to kill which tend to induce apoplexy in English teachers. They aren't very important here though, since the absence of a comma and inability to use a conjunction like and or but means you can't use any straightforward verb form like
He committed a crime and killed a bird.
or even its poetic/musical cousin
He committed a crime... [babybabyooh] killed a bird...
These kinds of questions are designed to force students to acquaint themselves with changing verbs around to make participles (verbal adjectives) and gerunds (verbal nouns).
A common focus of these kinds of questions is contrasting the active process of ing forms with the passive accomplished fact of ed forms. I am boring means I cause other people to feel boredom. I am bored means I feel boredom caused by someone or something else.
The passive ed form doesn't work because it has a direct object and kill doesn't have indirect objects. (Some other verbs like give or offer are different, since their indirect objects can be left where the direct object would normally go.)
The answer must be some form of killing, as either an adjective or a noun.
The most straightforward way this works is with a colon:
He committed a crime: killing a bird.
He committed a crime and that crime was the killing of a bird. This is almost exactly the same as the meaning of He committed a crime by killing a bird. If you were curious about the nuance, it's that by killing means that the action broke some law but it was possibly a broader statute with other aspects. The version with the colon means that the bird-killing is itself precisely the crime. The former could describe poaching or hunting out of season, but the latter (speaking strictly) couldn't unless more context was added. The bird-killing itself is a crime under any circumstances.
Another straightforward but separate way this works is with a comma:
He committed a crime, killing a bird.
He committed a crime and, as a result of that crime, a bird was killed.
Now, based on the way the question is being asked, you can't add either punctuation but
He committed a crime killing a bird.
can mean either one of them. Since it can also be understood as a terser form of he committed a crime while killing a bird, it can also mean that he was in the middle of the action of killing a bird (intentionally or not, connected or not) at the moment he committed the crime.
Now, absent any other context, most people are going to assume a connection between the crime and the ornithocide. That makes the grammatical ambiguity a little displeasing to start with, even if you don't take a very strict view about misplaced modifiers. Nonetheless, it's a perfectly valid answer to your teacher's question.
Also, unless your teacher told you only certain verb forms were allowed,
He committed a crime having killed a bird.
is also acceptable, although it suffers from the same ambiguity. Most people will take it as his action of having killed a bird constituted a crime but, grammatically, it could theoretically mean that, with the experience gained by killing a single bird, he proceeded to commit a crime.