The answer to your question about the apparently unwilling power of attorney may not be as simple as knowing the definition of "unwilling". It is a commonly found legal term
in documents in which an elder appoints another person, often an adult child, to do something: handle money or make healthcare decisions. The law allows the appointed person to decline the job of acting on the elder's behalf. Unwilling usually means "I don't want the job". After all, one can't be forced into being an agent on a power of attorney. It's a large responsibility. However, being "stubborn" is not the same thing the law normally means by unwilling. I am not sure if the person who is supposed to be the agent on the power of attorney just doesn't want to do the job, or doesn't want to bother to do the job properly.
First, it is important to have a conversation with the stubborn agent on the POA. If he/she is unable to do the caregiving job correctly and safely, there are likely others who are aware of this. If the agent is approached with an offer by the second appointed person to assume responsibility, with the support of those who are also aware of the problem, that's a start. One could say, for example, "Jane, I know taking care of mom is really difficult. We can all see how hard it is. I'm next in line to help, and I'd like to offer to take the burden off your shoulders and give you some relief." (Whatever your situation is, relate it to the agent). If that approach is used, it is not an accusation or a judgment, which should be avoided. It's just an offer to help.
If you, as the second on the POA, believe that the elder is getting inadequate care and that the elder may be neglected or endangered by the quality of care the current POA is providing, it's time to step up. Seek the advice of your elder's doctor and level with the doctor about your concerns. She may be able to guide you in suggesting that the current POA step aside and let you take over. You can use "doctor's orders" as leverage if appropriate.
You can also seek the advice of an elder law attorney who is familiar with caregiving issues. An attorney should know how to ask the right questions to find out if there is a real danger to the elder with the current POA's care. The law does have extreme measures available if a POA who is creating a dangerous situation for an elder outright refuses to allow the second in line POA to take over, to help, or to protect the elder. A court can order an evaluation, a change in caregiving, or as a last resort, a guardianship/conservatorship.
I recommend that the best approach to this problem is to start by offering to help, and ask if the second POA can step in and assist. If the first POA is willing to allow help, some responsibilities can be shared. If stubborn refusal puts the elder in continuing danger, do not hesitate to seek legal advice. When safety is at stake, one must act promptly to protect the elder.