Note:

>! This is not a popular way of aproaching this, and i personally love @Magma 's prove, yet I have to disagree.

Answer:

>! [![solution][1]][1]
>! The problem as stated is perfectly solveable, since nowhere in the rules it states, 
>! that you are limited by the borders of the 5x5 quare. So if you are to leave some hangover beyond the bounding box of the original shape, you are easily able to cover the original square, but you will have these ugly hangover parts left.
>! I do know that this is not a very elagant solution, but it is a valid one.
>!
>! But i do agree that the problem as it is (propably) meant to be solved is impossible.


  [1]: https://i.sstatic.net/KHUL2.png