I suppose if a young woman is leaving used sanitary pads on the bedroom floor, and not washing, then she might be depressed.
I suppose your DH does the whispering thing with her because he loves her and wants to have some communication with her, to help her, and if she whispers then he engages with her in that, because it's the communication she's giving and he wants to be on the receiving end of it, in case it's something significant about her feelings.
She is a grown woman almost, but individuals vary: some drop out of uni and jobs: some make unwise decisions. Parents will usually try to be there for them regardless. I expect your DH is very worried. He probably feels he doesn't have the relationship with his ex in which he can naturally and easily talk with her about their daughter and her mood. Worried about what to do, he probably is scared of losing his daughter in his life, if he should handle things wrongly. He probably doesn't know what to do, but he could ask a family therapist for advice.
Is she deliberately thinking, once a month: "I'm going to place this used sanitary towel facing upwards on the carpet", or is she just very absorbed in her feelings and careless and is just stripping off underwear and leaving it around - including the sanitary wear stuck to it? She sounds like she may be struggling with her mood and emotions.
Is there really no discussion with your DH? Are you expressing yourself as talking about your feelings as oppose to his daughter's revolting habits? "Are you saying: "Look, I'm very upset. Please listen and be supportive?". And does DH really just do the equivalent of going: "Shut up and leave it, will you!" "I don't want to hear what you think!"? when you are talking about your needs? If so, then other posters are right: you do have a problem with DH. He isn't listening or acknowledging your needs.
You don't - you can't expect a troubled teenager to have the same maturity, insight and empathy as a forty-something adult. You have to try to help her as that's what adults do with the young.
It's not a pleasant situation for you at all. But I suppose you have to be bigger than the situation and accept it's difficult. Say to DH it annoys you too much to pick up your step-daughter's mess and that DH will have to be the one to do it or to get her to do it.
Say to DH: " I feel that you are scared of losing her. I want to support and help you, because I can imagine how that must feel for you.".
I would never give the opinion that you should tell your DH that she shouldn't come to the house; that he should choose you over her, because I think that would cause real pain. You have annoyance and discomfort to put up with and frustration of working out how to deal with it, but you don't have real emotional agony. From what you write, your DH loves you - he just can't find a way to improve his daughter's worrying behaviour.
Without knowing the ins and outs, it would be very harsh to give the opinion: ' Just show your DH lots of views from outsiders agreeing with you that the girl is unacceptable and go on and on at him as often as you can until he cracks'.