Your question has several parts, and it should be addressed by an Illinois probate attorney, so this information is for general background purposes only.
First, the creation of a power of attorney instrument, whether durable, general, or special, relates to management of business affairs. It won't necessarily empower the named agent to "get Grandma out of the hospital" although the agent could sign financial/legal documents such as discharge papers or rental agreements for a new residence. Your grandmother probably needs a durable power of attorney for health care purposes, with an advance health care directive as part of it. That would be more akin to the right type of instrument to give medical care instructions.
Secondly, it seems to me that a conservatorship is more called for than a power of attorney instrument, in the facts you have described. You would have standing to seek such appointment, but so would her brother, and the court would have to decide between competing petitions. A sibling is higher in the statutes of some states providing priorities to petitioners for letters of conservatorships, than would a grandchild be.
You aren't going to have standing or capacity to block Grandma from signing a power of attorney instrument to your great-uncle, because she is presumed capable of signing that instrument, absent a decree determining her incapacity or adjudicating her a ward or conservatee. You could bring an action to invalidate it, but you would have a heavy proof burden to sustain.
Your best recourse is to consult with local counsel, who will probably recommend initiating a conservatorship or guardianship for her.
Good luck with it.
This is a complicated issue, and needs sorting out by an attorney in your state. Normally there is no limit to the numbers of people who can hold power of attorney. However, valid powers of attorney can only be granted by people who are mentally competent, which means they are mentally competent to make all decisions.
Please note that I do not practice in your state and that the above is not intended as legal advice, it is for educational purposes only. No attorney-client relationship is created or is intended to be created hereby. You should contact a local attorney to discuss and to obtain legal advice.
First, if she is "not all there" as you say, that suggests she does not have capacity to grant a power of attorney. If there is not already a power of attorney in place, then it may be necessary to obtain a court-ordered guardianship. If there are competing petitions, a court would review various factors to determine the best person to act.
If she does have capacity at any time to grant a power of attorney, the statutory durable power forms require the agent to act alone. Successors can be named. She can use a non-statutory power of attorney, but naming more than one individual could create serious complications.
A court-appointed guardian would have primary say as to residential placement, but court approval would be involved. An agent under a power of attorney would have similar authority, subject to being questioned if the actions are not in the best interest of the ward.
If your uncle does obtain responsibility, he would be under a duty to act appropriately, but you would want to intervene early if you run into any questions as it may become too costly and cumbersome, even if paid from your grandmother's funds, to litigate matters.
There are extremely serious considerations that cannot be covered within the scope of this site. The aboe is a very cursory examination based on limited information. You should consult an attorney.