Gertrude & The Beginning
Talk to either Shilop or Wilough and they will tell you about their secret playground but they will not give the location until they get something in return. Give them 100 coins.
Go back to Gertrude and talk to her. After that head for the Lumber Yard which is north-east of Varrock.
The Secret Playground (Lumber Yard)
Once you are at the Lumber Yard jump over the broken fence. You will see that many crates in the yard have cats inside them which are constantly *meowing* but none of them are Gertrude's cat. Head to the middle of the yard and climb up a ladder. You will find Gertrude's cat but you can't pick it up as it refuses to move. Use your Bucket of Milk on it. Try and pick it again. It will still refuse to move. Use your Raw Sardine on it. It dosen't want it. Head back to Gertrude's house.
Fluffs' Favourite Food
Talk to Gertrude and tell her that you have found her lost cat but it refuses to move. She will tell you that it loves Doogle Leaves and sardines. Head to the back of Gertrude's house and you will find a respawn of Doogle Leaves. Take one and use it with the Raw Sardine.
Go back to Gertrude's cat in the Lumber Yard. Use the Doogle Sardine with Gertrude's cat. Try and pick her up again and she still refuses to move. It says that she can hear kittens somewhere. Remember the meowing kittens in the crates earlier? Climb down the ladder and search the crates until you find a lovely kitten. Go back to Gertrude's Cat and use it with her. She run off back home to Gertrude's house. Return to Gertrude to claim your reward.