I'm really sorry to hear this. Situations like this are horrible, and they happen all too often. You're terrific to want to get your grandmother out of the mess they put her in.
If she is not competent, and goes in and out of lucidity, then she will not be able to do a power of attorney to you or even a will -- I don't know if she already has either?
However, in California, a legal guardianship for an adult is called a conservatorship and can only be established by an order of the probate court. A conservator is appointed for another adult when the probate court concludes that the adult, or conservatee, cannot manage his finances and personal affairs. This might be the best thing for her and for you. You'd be the guardian for her (conservator) and you would get to make decisions affecting your grandmother. You can have her where you think she needs to be, and you can request access to her accounts so you can handle her financial affairs as well. For her to get her property, she can do so, but if she has to sue for it, the conservator can sue for her if needed. You would need to ask for a few things -- to petition to be the personal conservator (daily living, long-range plans) and to be the conservator of the estate (financial matters). You could also petition for special powers for a person who may have dementia.
The only problem with this is that it is a public proceeding and your dad and uncles can challenge this, so you may be pitted against your family. That's up to you. There are other ways to go about this -- you can petition to get another person as a conservator for her and then you wouldn't be the conservator but it would be a neutral person who is sworn to look out for her best interests.
These are the alternatives that you have here.
If she is lucid at all, then she can give you power of attorney, but you will have to prove that she is lucid when she did this. She may be better off with conservatorship, which is what other people in this situation tend to do.