Question: How can we change the following sentence into negative one without changing the meaning? 

> 
> He did everything.
> 
> 
> 

Can we change it into this: 

> 
> He left nothing undone.
> 
> 
> 

Or 

> 
> Didn't he do anything 
> 
> 
>

Answer: 
> 
> He left nothing undone
> 
> 
> 

is pretty much the same as "he did everything". Your other example, however, is a question not a statement. 

Another possibility:

> 
> There was nothing he didn't do.
> 
> 
>
The question completely misunderstands itself.

'He did everything' is grammatically positive; semantically it means little. Define 'everything', please?

'He left nothing undone' is still positive, unless you think 'he ran away' or 'he surrendered' or 'he lost' grammatically negative. Do you?

'Didn't he do anything?' is a question; not comparable to the other examples.

'He did nothing' is still a positive statement. 'He did (whatever)' will always remain positive.

'He did not do anything' is as close as you could get to a negative statement.