Although the following example has two negatives, it's perfectly grammatical:
It's not true that "double negation" is ungrammatical.
It has two negators: the analytic negator not and the lexical negator un-. Despite this, the sentence is perfectly fine. So clearly, a rule stating that a sentence cannot contain two negations is simply false. We need a better rule:
It is usually considered non-standard to mark a single semantic negation multiple times.
Let's look at an example:
I ain't got no money. (non-standard English)
In this example, there is only one negation in terms of meaning, but it's been expressed multiple times, with both -n't and no. This is ungrammatical in Standard English, but it's perfectly grammatical in many non-standard dialects, such as AAVE (African American Vernacular English).
This language feature is called negative concord, and there's nothing intrinsically wrong with it; it's a feature in many other languages, such as Standard Italian. It just so happens that the dialects of English which are considered standard do not have this particular feature.
In Standard English, we would typically express the negation only once, like this:
I don't have any money.
Or like this:
I have no money.
In your example the negatives are not in concord. They're expressing two different negations in terms of meaning:
I won't stay here doing nothing.
And so your example is already fine in Standard English. Since both of the negators express distinct negations in terms of meaning, you can't remove either of them without changing the meaning of the sentence.