Personally I think so. I don't believe any animal should be kept a secret, especially if it is living with those who may not want it. Is it possible that the kitten may belong to someone else? Can you call any local animal shelters or check the newspaper in case someone has asked after a kitten matching the description of the one you found? I think it would be very responsible of you to ask around and see if the kitten has an owner
I'm sure your nana would appreciate it.
As much as anything else cats need exercise and that's usually achieved by letting the cat outdoors on a very regular basis, another thing some people have done (including my sister) is keep their cats as indoor cats - I don't feel that there is anything wrong with this as long as adequate space is provided for the animal to get as much exercise as it needs and wants, I do not honestly think such space would be able to be provided if the kitten is kept in secret.
There are also lots of other things you should consider such as a good quality food and plenty of water, a litter box and kitty litter (plus training the kitten to use said litter box), cleaning the litter box, vet bills (vaccinations, spaying/neutering, unexpected events should they ever occur) and toys/scratching posts.
I believe that while cats can be ideal companions for a single person, in a family home they really need to be a family pet - no animal stays secret forever. I really think it would be great if you had a look around to see if there are any "missing" posters or adverts around, you might find you reuinite the little kitten with a distraught owner.