My brother just lost his cat to cancer, he is alone, and really needs companionship of our kitten.
She uses her litter box and is eating good. He gave her a room with all her stuff in it. She doesn't come up to us yet.
My question is we both have bad knees so crawling around on the floor is not an option.
I have noticed when talking to her she will do that blinking thing with her eyes.
How can we encourage her to come up to us, if we're unable to get down on the floor with her?
First of all, get lots of cat furniture. Cats normally like perching up high, so if you give it places it can go to be higher off the ground, it will probably gravitate to them. You can make them even more enticing by placing them by windows or in rooms you spend a lot of time in. You can also place comfy beds on higher surfaces to encourage it to sleep there.
The other thing you can do is use treats or play to lead the cat to where you want it to go. Either can be done without bending over. With treats, you can toss them to the cat, and certain toys like a feather wand or laser pointer can be used while you are standing. This way you can get the cat to come up high, rather than having to bend over to interact.
Since it sounds like you're still in the introductory period of keeping the new cat to one room, that is actually a great time to get cat furniture. Place them all in its room, and encourage the cat to use them through play and treats. That way the furniture will have the cat's smell on them. Then once you are ready to introduce the cat to other areas of the house, move the furniture into those areas. This will spread its smell and make the introduction go more smoothly.
You mention you not being able to crawl, but are you able to just sit or lie on the floor, maybe propped up by a pillow? One way to convince a cat you're not a threat is to be in its space and then ignore it. The cat will eventually get bored and come out to investigate. However, the lower you are and the smaller you look, the less threatening you seem and the sooner their general fear will turn to boredom, hence why sitting or lying the floor (if you can do it) will work better than a chair or couch.
You can use that time to read a book, play games on your phone, talk softly, etc. Try not to approach the cat or make eye contact, loud noises or sudden large movements; they will interpret those things from strangers as possible threats and retreat/hide.
As I understand your question you wonder when the cat will start to interact with you, please correct me if I am wrong.
After a cat comes to a new place it will take some time before it feels safe, and the fact that there have been an other cat in the house will make it take a longer time before the new cat feel at home.
You will need to be patient the new cat will come to you the slow blink is a good sign and it shows the cat do respond in a positive way.
While you wait for the cat to trust you you can try to get the cat to play by using a toy on a stick, when a cat plays it will forget to be scared.
What has worked with me in the past is patting next to the place where I want them to come. At first they might ignore it, so you need to be patient. You have to repeat it over and over (not right away, but at different moments of the day for some weeks). Also try to avoid being forceful, just gently invite them to come.
For example when I want my cat to jump on the bed with me, I pat the bed a couple of times making a 'tap tap' sound. If she is distracted, I call her name, then I tap again.
Finally remember that cats need space to feel comfortable, so be sure to give them some room to relax.
TL;DR
Cats often need time to adjust to their new home and feel safe there. This is completely normal.
I second everyone who suggested getting cat furniture for her.
You can also get a cat toy (for example feathers on a long stick). If she chases the toy you can get her to jump up to you sitting on the sofa.
If you want to try it clicker training is an awesome activity that most cat-human-couples enjoy. If you can find any treat she especially likes (dry food, dried meat, etc) and pair it with a marker sound (the clicker) you can teach her a "recall". Or any other trick you both might enjoy. Positive interactions create a strong bond between an animal and their human companion.
