First,I want to say you are doing the best thing for your little girl by being home with her! Second, know that the fussiness will pass. Some days get REALLY long, but just remember in a few more months she will be past that super needy stage.
As for your husband, stop asking him or waiting for him to offer. Just delegate.
Try this and see if it works--
1.When he comes home from work, greet him with a big hug. Ask him how his day was, all that good stuff.
2.After he's been had a chance to get comfortable, hand your daughter to him and say,(pleasantly, of course) "Honey, I haven't had a shower all day, so I'm going to go take one. I'll be done in an hour."
3.Go take a shower. in an hour, go get your daughter from him.
4. Say, "I feel so refreshed now! Thanks babe, I really appreciate your help" (not sarcastically, really mean it)
5.THE MOST IMPORTANT THING--Do not rescue him or tell him he has to do something a certain way. If you leave him alone he'll figure it out!
When it comes to little babies, most men just need practice. Its important for us mommies to help them "practice" without butting in, even when it is so hard not too!
Also, there will be no more need to argue or talk about it--just simply tell him what you need-- I need 10 min. or I need 30 min.
At first, I would make the time 1 hour or less, until he gets comfortable by himself.
I think this method will keep you from dwelling on his lack of initiative (which is frustrating, I know) and also give you some much needed mommy time! "You can catch a fly better with honey than you can vinegar" :-)