What does こなしてみせます mean?

Question

次こそは... 完璧にこなしてみせます。

This is a speech from Shikamaru, a character from Naruto manga.

The nuance is, his very first mission as a leader ended in a giant failure with the rest of his team severely hurt and he only got a broken finger.

The above image is the moment that he vows to not fail again, which can be translated as something like:

Next time for sure... I'll carry out the mission perfectly!

But what makes me confused here is the phrase:

こなしてみせます
（熟して見せます?）

I was taught that, 物を見せる means show something (to somebody), but what does Shikamaru show here?

I have also looked up 熟す (こなす), which means mature, but こなしてみせます just does not make any sense to me.

Answer 1

I haven't looked up dictionaries, but inferring from the Japanese animations I have watched, こなす is something like doing something thoroughly and successfully. The most common form I have heard is 使いこなす (to use something in its full capability, or something like that). Never knew that it had a meaning like "mature"; never heard that kind of usage in animations.

I do not understand why you don't understand here みせる here. It means "show", as you said. So, he will show them his doing it thoroughly (or in current spoken English, he will show them himself doing it thoroughly) and successfully, next time.

Answer 2

For future readers who might be looking for answer, this is what you need:

https://www.kanshudo.com/grammar/%E3%81%A6%E3%81%BF%E3%81%9B%E3%82%8B

It's a grammar: てみせる, which means will definitely do ~

必ず期待にこたえてみせます。

I'll never fail to meet your expectations.

Also check out this answer: てみせる