Is it permissible for me to fast the five days that my deceased colleague missed from Ramadan, knowing that she is not one of my relatives?

The establishment of the obligation of fasting upon the deceased is summarized by their ability to make up the fasts and their failure to do so. If they died before being able to do so, then there is no obligation upon them, and no fasting is done on their behalf. As for one who dies with outstanding fasts, their guardian may fast on their behalf as a recommended act, and it is also permissible for a non-relative to fast on their behalf.