Thanks for your comments, everyone. You all gave some great answers. I've been thinking about my post as well. :-)
I have concluded that we do try our best every day. I have (frequently) had days when my best was to feed my family and make sure the dishes were clean that day. Period. I feel bad on those days because I know there's more that needs to be done. But for physical or emotional reasons, that is all I am capable of at that time. This is part of our mortality (our imperfections) and I know God understands this, and that if we truly strive to do better each day, then that's all we can do.
But what about problem-solving? Let's say (and this is not a true-life case) that I go to the bank and take out $100 to pay a bill that's due by the end of the week. For some reason I have to go home first, and then . . . you know how things go at home. I don't make it in time to pay the bill today. I will go tomorrow. So I leave the $100 on my dresser (which I would not do in real-life). Well, the next time I look for the money, the $100 is missing. I interrogate all my kids and my husband, but they all claim perfect innocence. (You had $100?!) Obviously this isn't ALL I can do to solve this problem.
So I clean out my purse; then I clean my dresser, look in the drawers, clean and organize the drawers, look behind and under the dresser (clean-up there too). No luck. I retrace my steps from the time I got home with the money yesterday—nothing. So now I look in the pockets of the clothes I was wearing yesterday. Then I make my bed and clean up my whole room. Still nothing. Have I done all I can?
Well, there's a good possibility that a child did something with the money, but was afraid to 'fess up. So now I ransack their rooms, cleaning as I go. After a few days I have torn apart and rebuilt the entire house, and the car too. I am praying the whole time, because I need to pay that bill. Have I now done everything I could do?
Well, I remember a rather large-ish hole in my living room wall (well, this part is true—we have a 3-inch hole in one of our walls). I suppose it is possible—although not likely—that a child could have shoved the money into that hole. I could tear away that entire section of wall to see if the money is in there. I consider this; I am capable of doing it, and repairing it again, too. But my feeling is, that it wouldn't really help. It's not likely the money is in there; it would be a lot of work, it would make a huge mess; so that's not an entirely logical solution to the problem. In addition, my first gut-feeling was not to do it.
So, NOW, do I pray and say "I have done everything I can do," or should I tear into the wall first? Have I really done everything, or am I just rationalizing? I have to say that I believe my first feelings (not to tear apart the wall) are probably correct. But sometimes I wonder.
As for asking for help, I know we need to. There are plenty of times that I know exactly what my kids want from me, but I wait for them to ask me first. Not because I am mean, but because sometimes they need to ask. I think there is something for us to gain by asking. And the scriptures tell us over and over again to ask God for what we want or need.
There was a family in our ward a while ago whose three-year-old was dying. He had so much internal bleeding (from an illness) that one evening his whole diaper was soaked in blood. His father gave him a blessing that he would live long enough to make it to the hospital. These parents and the baby's doctors had been treating the illness for several weeks; the parents had been praying for God's will to be done and that they could accept it. Everyone thought he would die. But that night, at the hospital, the mother had the thought that it was OK for her to ask for what she wanted. She wanted to keep her baby boy; she prayed for him to get well and live so she could keep him and raise him. He did heal and is completely cured; that was six years ago.
I believe there are times when God will just fill in the rest because we need Him to. But I also believe there are times we need to ask for His help. I guess the real answer is to keep trying, AND to keep asking. And to believe that the help, the solutions, whatever we need, will come. And to keep believing. That is the best we can do.