Important: In order to get Granny's final side mission, you MUST meet Slackjaw in Mission 3 and complete his missions.
Granny Rags is found in Mission 2. Once you get up to the area with a street and buildings, go to the far right corner and you should see her building. She will also be vocal when near her. It is very early on in the level. She has two side missions for you at this time. Complete them and she will be happy. Complete them and you won't see her again until late in Mission 7. While in that mission you will find a thug in the sewers on your normal route to escape the area. He will give you a key which you can use. At the end of this new route, you will find Granny Rags and Slackjaw.
WARNING SAVE YOUR GAME
Complete his mission and you will get the achievement. Reload your save and complete Slackjaw's mission instead to gain the related achievement for his side missions.