How much time must pass before I am not responsible for my aunt's debt and any house taxes?

Or are you talking about your Aunt going on Medicaid, in which case there might be a penalty or a lien if the house was sold below market rate?

Any profits from the sale of your Aunts house, which you bought, will go to her estate, and from her estate her debt/taxes will be paid. There is no reason for you to be responsible for any of her debt or her taxes. Now, if you had purchased her house way below market value, thus less profit money in her estate, there might be some debt/tax questions.

Did your Aunt leave you assets via a Will or a Trust? It depends on which document was used as to the amount of inheritance taxes you would need to pay. Your Aunt's attorney should advise you.