The best time to attract a cat is very late at night, when the world is at its most quiet. If you had her in the house, this would be easier! So I advise that you borrow or rent a humane trap so that you can do this in the house. The key word is patience, lots of patience.
Keep her in your room. You have to be very patient. Sit still in the room when the house is quiet. Keep the room very softly lit, not bright. A night light would be perfect. You can play music by Mozart very softly. It has a calming effect. (And it temporarily raises your IQ - according to some psychiatrists! ) Just sit in the same room and allow her to get used to your presence gradually. Even if she hides under the bed, continue doing that several times a day. If it's daytime, take your homework or a book with you. Just be with her often and let her decide when to move to another step. (If she has had a good home at one time, this stage might not have to last long; if she were a feral, you would probably have to do this for weeks.)
Put some very tasty food in that room. Do not attempt to touch her. She will decide when to allow that. After she is used to this situation, move the food bowl a bit closer, but not so close that she is frightened. When she seems to be comfortable with that, talk to her very softly, never making fast, sudden moves. Move the food dish a little bit closer, but don't rush things.
Eventually, try using a toy with several feet of length and and see if she will interact with you. (I don't think this will take long, because she is already responding to just a stick.) Continue that until she is no longer afraid to play or come near you. Don't make a move; let her make the decisions. You won't pet her. She will rub against you when she is ready. Wait until she starts rubbing against your hand on a regular basis--for some days. Never rush a stage.
The length of time this takes depends on whether she is a stray or a feral cat. It could take months for a feral. If she was a pet, who knows? It could be only days. Time will tell. From what I understand, the time varies drastically. I My guess is that she had a home at one time. I hope she soon makes friends. Keep us posted, please.
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A dog, I have always said, is prose; a cat is a poem. ~Jean Burden