A durable power of attorney is possible under a specific state law. I assume Washington, to give a concrete jurisdiction. A power of attorney imposes a certain duty on the agent, basically to act in the interest of the principle. RCW 11.125.400 specifically addresses health care authority. No aspect of the law requires an agent to "take care of the principal", so declining to give said care is not a breach of the agent's duty. It does not trigger automatic revocation of the POA. RCW 11.125.180 says how the agent can resign. The closest you can come is that an agent can resign
by giving notice to the principal and, if the principal is
incapacitated: (1) To the conservator or guardian, if one has been
appointed for the principal, and a coagent or successor agent, if
designated; or (2) If there is no person described in subsection (1)
of this section: (a) To any person reasonably believed by the agent to
have sufficient interest in the principal's welfare
but this has not been done.
It is not clearly established that the agent lied, so we will have to make up facts in order to manufacture a legal wrong (and if we can't cook up a legal wrong, there is none). There is no contractual obligation between the agent and the care-provider, so no breach of contract. The one imaginable legal path for going after the agent is fraud, which is obtaining something of value by proffering a knowingly-false material statement. The thing of value would, presumably, be "admission to the hospital". The statement would be material if the hospital would not have admitted the principal had they known that the statement was false.
There would be difficulties in proving the materiality of the statement. Here is a transcript of the conversation between the hospital and the agent:
Doctor: "I feel this is best done as an out patient. But if you can't
take care of her then I will admit her".
Agent: "I can't, please admit her".
A reasonable interpretation of the response is that "I'm busy, it would be inconvenient, I just can't", which is presumably true. If the doctor has a different standard of "can't" in mind – some line that he can't cross – it is up to him to make that clear. In other words, "can't" typically is used to say "won't", and it does not mean "have no physical or financial capacity to do so".
An alternative transcript would be:
Doctor: "We absolutely cannot admit her, unless you will be in another state during this period and have no person that you can rely on to take care of her in your stead.
Agent: "I will be in Florida for the whole month, there is no other person who can take care of her".
(but Agent is not going anywhere, he just doesn't wanna, and there are plenty of family members around who would step in). While this could be a material false statement, it depends on there being completely unrealistic demands from the hospital.
This is not a possible criminal matter in Washington, it is a civil matter, in which case the hospital would have to prove that it suffered harm by admitting the principal under those conditions. They get paid one way or another, so establishing more than $1 of damages could be challenging. The most likely path for substantial damages would involve the insurance company.