Is it permissible for my husband and me to stay overnight at the home of my husband's maternal aunt, whose husband has passed away, and who lives with her an immodestly dressed daughter, knowing that the gatherings may involve much laughter and joking, and that my husband participates in them?

If a woman who is displaying her adornments is not a mahram (definition) to your husband, then she is a stranger (ajnabiyyah) to him. He must adhere to the Shari'ah guidelines, such as lowering his gaze and not being in seclusion with her. She, in turn, must observe hijab (definition) and not speak in a soft, alluring manner. You should advise her and her mother. There is no harm in your husband staying overnight at his aunt's house without mingling with her daughter if the house is spacious enough for that, and if you know that the household members are genuinely pleased with it. Otherwise, he should not, because what is taken out of shyness is like what is taken by force.