Will I be responsible for medical medicare hospital bills if my mother is unable to cover them with her assets?

Your mother's estate is respondible for her medical bills, not you. Should your mother be admitted to a skilled nursing or assisted living facility, cross out "personal representative" wherever it appears and write in "agent under Durable Power of Attorney" or "DPOA agent" or, if there is no Durable Power of Attorney, "next of kin."  The phrase "personal representative" makes you personally responsible for your mother's bills.  It is illegal for a nursing home which accepts Medicare or Medicaid to suggest or require a third party guaranty of payment but it is still common.