S.B.
That pus pocket is called an absess.
It means infection.
I just went through the whole absess thing, so trust me.
Since she is 5, the simplest thing would be to get that dead tooth out so the infection can drain. She may even need antibiotics.
Have her rinse with warm salt water. Try a warm compress. Quit pushing on it. Get her in to see a dentist.
Best wishes.