The answer really depends on your girlfriend's ability to understand what a power of attorney is and what it means in light of her medical condition and the fact that she is sedated. You should talk to her doctors (and I know this is tricky because without a power of attorney they likely will not tell you what is going on) and ask them if she has the mental capacity to understand what a power of attorney is and what it does as well as knowingly and voluntarily sign it.
Typically, hospital staff will agree to be present and 'witness' a power of attorney. Most hospitals have a policy on whether or not they can sign a power of attorney (in light of the patient's mental capacity). In Indiana though, you will also need a notary there to witness her signing the power of attorney and notarize it - this is also a service many hospitals provide. Either way, you may want to contact an attorney just to make sure there are no problems with the power of attorney and that you understand how to use it so there are no problems for you as well.
I wish you and your girlfriend the best and hope she recovers and is back home soon.
While I appreciate the opportunity to answer your question, my thoughts and impressions of your case may change given more information. Typically, I would give advice following a face-to-face meeting where we could both ask questions of each other about your situation. Therefore, my answer is intended to be general guidance and not legal advise for you to rely on in your particular situation. Moreover, my response to you is not intended to establish an attorney-client relationship. If you would like to hire me as an attorney, you must contact my office and we must enter into an agreement for me to represent you.
Only if she is competent to sign a legal document at the time of signing.
If she will never be competent-a court appointed guardian would be the answer.
The answer given does not imply that an attorney-client relationship has been established and your best course of action is to have legal representation in this matter.