As financial POA am I responsible for my mothers bill at a SNF?

No! It is not your Bill!

Contact a ombudsman at the office of aging in your area.
They will lead you to the correct resources.

But you are not responsible for her medical bills.

Lots of questions
Did you sign something that makes you liable to pay anything?
What exactly are those $10,000 charges.
Medicaid never pays for Cable, Internet/Phone.
Medicaid will pay for her care minus her monthly SSDI or other social security monthly payment, which is labeled as patient/resident contribution.

The County that encapsulates the town, in which the facility sits, calculates her payment due each month, Why? b/c each U.S. state allows something small $50-84/month for the resident's personal use, that amount is determined at the county level.

In my family members case SSDI missed paying 2 months, due to a closed bank account, we couldn't get it changed in time to get the direct deposit routed to the new bank account, 3 years later, crickets chirping is the only response we hear.

Her facility might be charging her account for incidentals, activities and hair salon things

Which means you'll need an invoice that outlines the $10,000 charges. That ONLY your mother will be liable to pay. (If that contract doesn't stipulate you're liable).
Usually surprise bills are incidentals that have been accumulating since Day 1.
a sudden $10,000 sounds like a error.
Don't let them bully you. Our horrific social worker will verbally demand immediate payments. with threats of evictions within 24 hours.
Their unethical tactics are aggressively creative.
Illegal, but their thinking = laws are only meant for those who get caught

All in all, you'll find answers when you contact the billing department, and when you re-examine the contract.

You are not financially responsible personally for any of your mother's expenses. You're responsible to pay for her expenses out of her bank accounts or by selling her assets. If she's out of money, then the nursing home will try to intimidate you with a shakedown and threaten that you're personally responsible for paying her bills out of your own money. You're not and don't let them shake you up because they will try to. Your mother owes ABSOLUTELY NOTHING after Medicaid paid. Nursing homes are allowed to charge Medicare/Medicaid a different price than if you're private paying in cash. They got paid by Medicaid, but they still want that extra 10 thousand. Call her Medicaid caseworker (yes she has one from the state) and ask them to send you the spreadsheet of when she was in the Medicaid "spend down" period and what months she was paid by Medicaid. Normally people on Medicaid and SNAP get these forms periodically. If you have not them request one. If Medicaid already paid mom's monthly bill, and sometimes they pay retro-actively then the nursing home is owed nothing from private pay.

Is her social security being paid to the nursing home? That could be what they are referring to. She has a share of costs and you are responsible to ensure that is being paid for your mom from her retirement benefits.

Who filed for Medicaid? You should go have a sit down meeting with the finance department and get a full understanding of what is going on and how this all works.

Edit: posted before I read other responses. We are all on the same page. Can you let us know if you have been paying her SS to the NH?

I am assuming that you understand how to sign as POA. If not, that is easily found out on google or another search engine.
Your Mother's estate is responsible for the bill. If your mother has no funds, which is likely as she is on medicaid, then there is no money in the estate to pay this bill. Can you tell us whether a facility is insisting that you personally pay this bill? Have you been in discussion with the facility?

No, not you personally. Your mother's money pays for her care. So all of your mother's income must go to the NH, except a small amount for personal needs.

No, unless you signed paperwork as being responsible for payment. You should sign everything with POA at the end of your name.

Your Mom has Medicaid paying for her care? Are you using her Social Security and any pension she receives to pay her share? Because in the Medicaid application her SS and pension are worked into the formula. Lets say the cost per month for Moms care is 8k. Her SS and/or pension amounts to 2k monthly, that money goes toward her care (less monies allowed in Personal needs acct) with Medicaid paying 6k for her care. If you have not been paying her share, then that is why you have a balance. Once on Medicaid, none of her income can be spent on anything, like a house. Only to offset the cost of her care.