Medicare Part Ahospital insurance usually leaves some chunk of hospital bills unpaid, even with supplemental Medigap insurance like your mother has. Many hospitals will negotiate with low-income Medicare patients about the part of the bill that Medicare and insurance don't pay. Sometimes a hospital will waive the entire unpaid balance, as the Elk Grove hospital did. But a hospital is not legally obligated to do so. Each hospital makes its own rules about how much of personal bills they will try to collect and how much they will let go. (One of the things some hospitals sometimes consider is total "household" income.) So, the Los Gatos hospital has the legal right to ask your mother for payment of the $1,024.
They do not have the right, however, to hold you personally responsible for the bill. Having either power of attorney for finances or power of attorney for health care does not make you personally responsible for any of your mother's unpaid bills. It does, though, give you the right to negotiate with the hospital on your mother's behalf. Contact the billing office of the hospital and ask to discuss the bill with them. Explain your mother's financial situation, including the fact that her income is low enough for her to qualify for assistance with the Medicare prescription drug program. Ask if they will waive the unpaid bill, or at least accept a lower amount.
If you don't have any luck directly with the hospital, you might contact the Health Insurance Counseling and Advocacy Program (HICAP), whose free, expert counselors may be able to help you negotiate with the hospital. You can find the local HICAP office nearest you by going to the list provided online by the state Department of Aging or by calling the central HICAP phone number 800-434-0222.
For future medical care, you might want to consider applying for Medi-Cal (California's version of Medicaid) for your mother. Medi-Cal is for people with low income and few assets, and in California eligibility can be extended to people who might have slightly too much income if they also have have substantial medical bills. If your mother qualifies, Medi-Cal can pay almost all portions of her future hospital or doctor bills that Medicare and insurance don't cover. To find out about Medi-Cal, go the Medi-Cal "Where to Get Help" page on the web site of the California Department of Health Care Services.