Most cities have a Legal Aid Society that provide free advice in just such situations. You may also call the nearest law school and see if they have a program in which law students provide help (under supervision of a licensed attorney). If your family member isn't competent (sounds like the case here), getting a POA will require you to petition the court. You'll have to provide proof of his incompetence. Most likely, you will need to start at your local probate court. Sometimes, they make it easier by providing forms and instructions.
Don't rely on what people on YA tell you. You don't know the people advising you, or whether they know what they are talking about. Talk to the clerk in the probate court, and/or get a lawyer. Just as an aside, a lawyer at a general practice can probably do this, as could a family law specialist. It might not cost as much as you think, because it may not require a court appearance. Find a lawyer that offers free consultations. They can briefly discuss what you need and provide you with an estimate.
>Is it true in Japan that parents can put their children in a orphanage-like institute
Parents can put their children in an institute like that if they can't or do not want to raise children for various reasons like financial and others.
>but as long as they do not waives their parental rights, the child can't be adopted and force to stay there?
Yes, of course. They still have the custody of children.
>If true, what is the age a child is no longer adhere to this law?
The age of adulthood is 20 in Japan. But you need to be more specific which law you are talking about. In fact, you can't adopt a child without the consent of real parents no matter how old the child is.
>And does the law and age varies from prefectures to prefectures?