Here is the fragment again:
Including the frying pan with encrusted egg and the greasy silverware full
of fingerprints.
You correctly identified the fragment as an afterthought phrase. Including, the transition that begins the fragment, is the clue.
To fix this fragment, you can modify the information so that you have a main clause. Here is the correction:
Diana considered a frying
pan with encrusted egg and greasy silverware full of fingerprints
as clean.
Or you could use Punctuation Rule 7:
Main Clause + , + Afterthought Transition + Ø + Details.
The correction would look like this:
Diana put away what she considered clean dishes,
including the frying pan with encrusted egg and the greasy
silverware full of fingerprints.
Enjoy the respect that you have earned in the kingdom!