Youve missed to top of your code
The easiest way to put a repeat into a game once it is already working is to move the whole game into a function, then add your loop that askes the user if they want to repeat in main and call the function. A do ... while loop works well for this since you always want to play the first game.
I'm slightly surprised that the loop at line 31 doesn't just do into an infinite loop when you guess a letter that is in the word. Line 34 - 36 actually replace a single correctly guessed letter in the word and are only done once for each letter guessed.
You code could use a proper refactor it is messy and confusing. I can tell this because the lines
//Prompt the user to guess a character in the word.
cout << "(Guess) Enter a letter in word " << word << " > ";
cin >> letter;
Appear 3 times in your code. This should not be required, this program can easil be written with the guess request in only 1 place (I would say just inside the do ... while loop. The only other loop required would be to look for multiple occurances of the same letter that was guessed.