Normally people would say, “I need to go to rehab.” “Rehabilitation” is correct, but people normally shorten the word to “rehab.”

Say either "Now I have injured" or "I injured" or "I have injured". Don't say "now I injured" because the past tense conflicts with "now". This is grammatical but not clear in the context: "I've been to the hospital". Are you in the hospital? Did you go to the hospital? Either say "so now I am in the hospital" or "so I went to the hospital" or "so I have been IN the hospital (for a while)". "Rehabilitation" is a commonly used term, but "gotta" is slang.

You could also use the word "rehab", which would be a more informal option. "I started rehab for my knee injury this week"

Yes, you can definitely use the word "rehabilitation" in the context of your knee injury and recovery. Rehabilitation refers to the process of restoring or improving physical strength, function, and mobility after an injury or surgery. It often involves exercises, therapies, and other treatments designed to aid in the recovery and healing of the injured area. When you say, "I gotta go to rehabilitation," it conveys that you need to attend sessions or appointments where you will receive specific treatments or exercises to aid in the recovery of your injured knee. This phrase indicates your commitment to the rehabilitation process and your understanding of the importance of seeking professional help to regain your knee's functionality.

I would use the word physiotherapy.