public boolean endsGerundIgnoreCase(String str)

Given a String, return true if it ends with "ing". false if it doesn't.
The method is case-INSENSITIVE, that is, capitals (upper case letters) and lower case letters match.

Explanation:
This can be done with just one easy change:

public boolean endsGerundIgnoreCase(String str) {
  boolean ends = false;
  str = str.toLowerCase(); 
  if ( str.endsWith("ing") ) {
    ends = true;
  }
  return ends;
}

(*) Use toLowerCase(), which creates a version of the string where all letters are lower case.

Note that you need to RE-ASSIGN the value of this new string to the str variable:

str = str.toLowerCase();