litter training for an older cat

Question
A spayed, female cat (approx. 3 year old) adopted us and now spends much of her time indoors but still prefers to be outdoors in warmer weather. We are trying, unsuccessfully, to introduce her to a litter box....so far, when we put her in the box she paws the litter and lies down in it. She has not "produced" anything to date. Is it too late for her to catch on or is there some trick we might try? There are coyotes in our area ( new phenomenon around here) and we hate the idea of her being outdoors at night. If she goes to sleep with us she asks to go out around 3 or 4 am. Any suggestions would be greatly appreciated. Thanks.

Answer
Hello Noni,
Litter box habits are usually instinctual as well as reinforced by the mother kitty very early on.  What happens if you don't let her out in the early morning?  Has she gone on the floor?  You might try putting her and her litter box in the bathroom, mudroom, or somewhere where she is fairly confined, and will have no choice but to use the box.  Hopefully, instinct will kick in when she has no other choice.
I hope this will be of help.
Thank you again for your inquiry.
Tomay