Here is the item again:
Hissing and spitting like an angry alligator caught by its tail.
You correctly identified the fragment as a participle phrase. Hissing, the present participle that begins the phrase, is the clue.
To fix this fragment, you could add a subject and verb, like this:
The old coffee maker was
hissing and spitting like an angry alligator caught
by its tail.
Or you could use Punctuation Rule 3:
Main clause + , + participle phrase.
The correction would look like this:
The old coffee maker brewed Christine's breakfast,
hissing and spitting like an angry alligator caught
by its tail.
Or you could use Punctuation Rule 4:
Participle phrase + , + main clause.
And the correction would look like this:
Hissing and spitting like an angry alligator caught
by its tail, the old coffee maker brewed Christine's
breakfast.
Enjoy the honor!