stroke
Going through the same process you are, but several days behind you. My sister had a stroke Thursday evening and is still being evaluated at the local hospital. She is alive and can move all four limbs, knows her name, but has aphasia and can't find the words to finish a sentence. She said today, "I want..." and that was it. It's hard because only one person can visit due to covid.
I did give her a notebook and a pen for when she can progress in her communication a bit more, so do give her one and several pens when she gets settled in. She may not be able to use it at first, but you will have it to write notes and questions to the doctor/nurse when you go up to get her if they allow that or for when she's moved in to the facility near you. You can prop that notebook up on her bedside table for them to see along with your phone number.
My son did say that one bit of advice that he learned when I was incommunicado in the hospital about eight years ago is that doctors all have their own opinions about longterm outcome. Listen, but don't get flustered by all the different opinions. There is no real timeline for each patient. One week and she's being moved into rehab sounds like real progress. John Hopkins stroke questions and timeline is a good reference online.
As far as finding a rehab center near you if you haven't found one, call the hospital resource person. They will be able to look for a center that deals with stroke patients that is near you. If there is no resource person, which is unlikely, then you'll have to call the nearest nursing facility and tell them that you are looking for one that has a stroke rehabilitation program. They should be able to tell you which places offer that kind of service. (But it kind of sounds like you have found one already.)
Keep in touch if you want and we can compare notes on their recoveries and our frustrations dealing with medical facilities.