How do I get medical power of attorney over my dad in the state of Illinois?

According to the medicare &You hand book, certain people with disabilities may apply for medicare if under the age of 65. However, to receive this the disability must be evaluated and approved by the Social Security medical personnel.
I would start with your PCP to establish the disability. Then contact Social Security office for further instructions and requirements.
I hope this helps you some.
I do believe that medicaid will still become involved

Your Dad is not old enough for Medicare. Cannot get till 65. Is he on SS? Old enough for that if 62 but reduced because he isn't waiting till full retirement at 66. If on SS he will get Medicare automatically at age 65. Medicaid should be paying for his care and any medical till then.

I have never heard of a court appted POA. POA is usually assigned. You may have to get guardianship. Check with your state laws.

Power of Attorney is not over someone. POA is given to a person to act behalf of an individual. If your father is still able to make decisions, he will have to assign you as his POA. If he is not able to make his own decisions, you will have to apply to the court to be appointed his Power of Attorney for medical and financial. You should consult an attorney who specializes in elder issues.