Is it permissible to complete the fast of the Day of Ashura for someone who did not intend to fast from the night before, knowing that they have not eaten or drunk anything after becoming aware of the day?

You should complete the fast of the day you were reminded, as long as you have not done anything that invalidates the fast; for the majority of scholars, it is not obligatory to form the intention the night before for a voluntary fast. As for the day before, you are not considered to have fasted it due to the absence of the intention to fast, and fasting is not valid with a delayed intention. It is recommended for you to fast the eleventh day along with the tenth, in order to differ from the Jews.